After reading chapter 6, analyze the structure of advanced encryption standards and why it makes it so strong. The initial post must be completed by wednesday at 11:59 eastern. You are also required to post a response to a minimum of two other students in the class by the end of the week. You must use at least one scholarly resource. Every discussion posting must be properly APA formatted.

CRYPTOGRAPHY AND NETWORK SECURITY PRINCIPLES AND PRACTICE SEVENTH EDITION GLOBAL EDITION

William Stallings

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3

CONTENTS Notation 10

Preface 12

About the Author 18

PART ONE: BACKGROUND 19

Chapter 1 Computer and Network Security Concepts 19

1.1 Computer Security Concepts 21 1.2 The OSI Security Architecture 26 1.3 Security Attacks 27 1.4 Security Services 29 1.5 Security Mechanisms 32 1.6 Fundamental Security Design Principles 34 1.7 Attack Surfaces and Attack Trees 37 1.8 A Model for Network Security 41 1.9 Standards 43 1.10 Key Terms, Review Questions, and Problems 44

Chapter 2 Introduction to Number Theory 46

2.1 Divisibility and the Division Algorithm 47 2.2 The Euclidean Algorithm 49 2.3 Modular Arithmetic 53 2.4 Prime Numbers 61 2.5 Fermat’s and Euler’s Theorems 64 2.6 Testing for Primality 68 2.7 The Chinese Remainder Theorem 71 2.8 Discrete Logarithms 73 2.9 Key Terms, Review Questions, and Problems 78 Appendix 2A The Meaning of Mod 82

PART TWO: SYMMETRIC CIPHERS 85

Chapter 3 Classical Encryption Techniques 85

3.1 Symmetric Cipher Model 86 3.2 Substitution Techniques 92 3.3 Transposition Techniques 107 3.4 Rotor Machines 108 3.5 Steganography 110 3.6 Key Terms, Review Questions, and Problems 112

Chapter 4 Block Ciphers and the Data Encryption Standard 118

4.1 Traditional Block Cipher Structure 119 4.2 The Data Encryption Standard 129 4.3 A DES Example 131 4.4 The Strength of DES 134

4 CONTENTS

4.5 Block Cipher Design Principles 135 4.6 Key Terms, Review Questions, and Problems 137

Chapter 5 Finite Fields 141

5.1 Groups 143 5.2 Rings 145 5.3 Fields 146 5.4 Finite Fields of the Form GF(p) 147 5.5 Polynomial Arithmetic 151 5.6 Finite Fields of the Form GF(2n) 157 5.7 Key Terms, Review Questions, and Problems 169

Chapter 6 Advanced Encryption Standard 171

6.1 Finite Field Arithmetic 172 6.2 AES Structure 174 6.3 AES Transformation Functions 179 6.4 AES Key Expansion 190 6.5 An AES Example 193 6.6 AES Implementation 197 6.7 Key Terms, Review Questions, and Problems 202 Appendix 6A Polynomials with Coefficients in GF(28) 203

Chapter 7 Block Cipher Operation 207

7.1 Multiple Encryption and Triple DES 208 7.2 Electronic Codebook 213 7.3 Cipher Block Chaining Mode 216 7.4 Cipher Feedback Mode 218 7.5 Output Feedback Mode 220 7.6 Counter Mode 222 7.7 XTS-AES Mode for Block-Oriented Storage Devices 224 7.8 Format-Preserving Encryption 231 7.9 Key Terms, Review Questions, and Problems 245

Chapter 8 Random Bit Generation and Stream Ciphers 250

8.1 Principles of Pseudorandom Number Generation 252 8.2 Pseudorandom Number Generators 258 8.3 Pseudorandom Number Generation Using a Block Cipher 261 8.4 Stream Ciphers 267 8.5 RC4 269 8.6 True Random Number Generators 271 8.7 Key Terms, Review Questions, and Problems 280

PART THREE: ASYMMETRIC CIPHERS 283

Chapter 9 Public-Key Cryptography and RSA 283

9.1 Principles of Public-Key Cryptosystems 285 9.2 The RSA Algorithm 294 9.3 Key Terms, Review Questions, and Problems 308

CONTENTS 5

Chapter 10 Other Public-Key Cryptosystems 313

10.1 Diffie-Hellman Key Exchange 314 10.2 Elgamal Cryptographic System 318 10.3 Elliptic Curve Arithmetic 321 10.4 Elliptic Curve Cryptography 330 10.5 Pseudorandom Number Generation Based on an Asymmetric Cipher 334 10.6 Key Terms, Review Questions, and Problems 336

PART FOUR: CRYPTOGRAPHIC DATA INTEGRITY ALGORITHMS 339

Chapter 11 Cryptographic Hash Functions 339

11.1 Applications of Cryptographic Hash Functions 341 11.2 Two Simple Hash Functions 346 11.3 Requirements and Security 348 11.4 Hash Functions Based on Cipher Block Chaining 354 11.5 Secure Hash Algorithm (SHA) 355 11.6 SHA-3 365 11.7 Key Terms, Review Questions, and Problems 377

Chapter 12 Message Authentication Codes 381

12.1 Message Authentication Requirements 382 12.2 Message Authentication Functions 383 12.3 Requirements for Message Authentication Codes 391 12.4 Security of MACs 393 12.5 MACs Based on Hash Functions: HMAC 394 12.6 MACs Based on Block Ciphers: DAA and CMAC 399 12.7 Authenticated Encryption: CCM and GCM 402 12.8 Key Wrapping 408 12.9 Pseudorandom Number Generation Using Hash Functions and MACs 413 12.10 Key Terms, Review Questions, and Problems 416

Chapter 13 Digital Signatures 419

13.1 Digital Signatures 421 13.2 Elgamal Digital Signature Scheme 424 13.3 Schnorr Digital Signature Scheme 425 13.4 NIST Digital Signature Algorithm 426 13.5 Elliptic Curve Digital Signature Algorithm 430 13.6 RSA-PSS Digital Signature Algorithm 433 13.7 Key Terms, Review Questions, and Problems 438

PART FIVE: MUTUAL TRUST 441

Chapter 14 Key Management and Distribution 441

14.1 Symmetric Key Distribution Using Symmetric Encryption 442 14.2 Symmetric Key Distribution Using Asymmetric Encryption 451 14.3 Distribution of Public Keys 454 14.4 X.509 Certificates 459

6 CONTENTS

14.5 Public-Key Infrastructure 467 14.6 Key Terms, Review Questions, and Problems 469

Chapter 15 User Authentication 473

15.1 Remote User-Authentication Principles 474 15.2 Remote User-Authentication Using Symmetric Encryption 478 15.3 Kerberos 482 15.4 Remote User-Authentication Using Asymmetric Encryption 500 15.5 Federated Identity Management 502 15.6 Personal Identity Verification 508 15.7 Key Terms, Review Questions, and Problems 515

PART SIX: NETWORK AND INTERNET SECURITY 519

Chapter 16 Network Access Control and Cloud Security 519

16.1 Network Access Control 520 16.2 Extensible Authentication Protocol 523 16.3 IEEE 802.1X Port-Based Network Access Control 527 16.4 Cloud Computing 529 16.5 Cloud Security Risks and Countermeasures 535 16.6 Data Protection in the Cloud 537 16.7 Cloud Security as a Service 541 16.8 Addressing Cloud Computing Security Concerns 544 16.9 Key Terms, Review Questions, and Problems 545

Chapter 17 Transport-Level Security 546

17.1 Web Security Considerations 547 17.2 Transport Layer Security 549 17.3 HTTPS 566 17.4 Secure Shell (SSH) 567 17.5 Key Terms, Review Questions, and Problems 579

Chapter 18 Wireless Network Security 581

18.1 Wireless Security 582 18.2 Mobile Device Security 585 18.3 IEEE 802.11 Wireless LAN Overview 589 18.4 IEEE 802.11i Wireless LAN Security 595 18.5 Key Terms, Review Questions, and Problems 610

Chapter 19 Electronic Mail Security 612

19.1 Internet Mail Architecture 613 19.2 Email Formats 617 19.3 Email Threats and Comprehensive Email Security 625 19.4 S/MIME 627 19.5 Pretty Good Privacy 638 19.6 DNSSEC 639 19.7 DNS-Based Authentication of Named Entities 643 19.8 Sender Policy Framework 645 19.9 DomainKeys Identified Mail 648

CONTENTS 7

19.10 Domain-Based Message Authentication, Reporting, and Conformance 654 19.11 Key Terms, Review Questions, and Problems 659

Chapter 20 IP Security 661

20.1 IP Security Overview 662 20.2 IP Security Policy 668 20.3 Encapsulating Security Payload 673 20.4 Combining Security Associations 681 20.5 Internet Key Exchange 684 20.6 Cryptographic Suites 692 20.7 Key Terms, Review Questions, and Problems 694

APPENDICES 696

Appendix A Projects for Teaching Cryptography and Network Security 696

A.1 Sage Computer Algebra Projects 697 A.2 Hacking Project 698 A.3 Block Cipher Projects 699 A.4 Laboratory Exercises 699 A.5 Research Projects 699 A.6 Programming Projects 700 A.7 Practical Security Assessments 700 A.8 Firewall Projects 701 A.9 Case Studies 701 A.10 Writing Assignments 701 A.11 Reading/Report Assignments 702 A.12 Discussion Topics 702

Appendix B Sage Examples 703

B.1 Linear Algebra and Matrix Functionality 704 B.2 Chapter 2: Number Theory 705 B.3 Chapter 3: Classical Encryption 710 B.4 Chapter 4: Block Ciphers and the Data Encryption Standard 713 B.5 Chapter 5: Basic Concepts in Number Theory and Finite Fields 717 B.6 Chapter 6: Advanced Encryption Standard 724 B.7 Chapter 8: Pseudorandom Number Generation and Stream Ciphers 729 B.8 Chapter 9: Public-Key Cryptography and RSA 731 B.9 Chapter 10: Other Public-Key Cryptosystems 734 B.10 Chapter 11: Cryptographic Hash Functions 739 B.11 Chapter 13: Digital Signatures 741

References 744

Credits 753

Index 754

8 CONTENTS

ONLINE CHAPTERS AND APPENDICES1

PART SEVEN: SYSTEM SECURITY

Chapter 21 Malicious Software

21.1 Types of Malicious Software (Malware) 21.2 Advanced Persistent Threat 21.3 Propagation—Infected Content—Viruses 21.4 Propagation—Vulnerability Exploit—Worms 21.5 Propagation—Social Engineering—Spam E-mail, Trojans 21.6 Payload—System Corruption 21.7 Payload—Attack Agent—Zombie, Bots 21.8 Payload—Information Theft—Keyloggers, Phishing, Spyware 21.9 Payload—Stealthing—Backdoors, Rootkits 21.10 Countermeasures 21.11 Distributed Denial of Service Attacks 21.12 References 21.13 Key Terms, Review Questions, and Problems

Chapter 22 Intruders

22.1 Intruders 22.2 Intrusion Detection 22.3 Password Management 22.4 References 22.5 Key Terms, Review Questions, and Problems

Chapter 23 Firewalls

23.1 The Need for Firewalls 23.2 Firewall Characteristics and Access Policy 23.3 Types of Firewalls 23.4 Firewall Basing 23.5 Firewall Location and Configurations 23.6 References 23.7 Key Terms, Review Questions, and Problems

PART EIGHT: LEGAL AND ETHICAL ISSUES

Chapter 24 Legal and Ethical Aspects

24.1 Cybercrime and Computer Crime 24.2 Intellectual Property 24.3 Privacy 24.4 Ethical Issues 24.5 Recommended Reading 24.6 References 24.7 Key Terms, Review Questions, and Problems 24.A Information Privacy

1Online chapters, appendices, and other documents are at the Companion Website, available via the access card at the front of this book.

CONTENTS 9

Appendix C Sage Exercises

Appendix D Standards and Standard-Setting Organizations

Appendix E Basic Concepts from Linear Algebra

Appendix F Measures of Secrecy and Security

Appendix G Simplified DES

Appendix H Evaluation Criteria for AES

Appendix I Simplified AES

Appendix J The Knapsack Algorithm

Appendix K Proof of the Digital Signature Algorithm

Appendix L TCP/IP and OSI

Appendix M Java Cryptographic APIs

Appendix N MD5 Hash Function

Appendix O Data Compression Using ZIP

Appendix P PGP

Appendix Q The International Reference Alphabet

Appendix R Proof of the RSA Algorithm

Appendix S Data Encryption Standard

Appendix T Kerberos Encryption Techniques

Appendix U Mathematical Basis of the Birthday Attack

Appendix V Evaluation Criteria for SHA-3

Appendix W The Complexity of Algorithms

Appendix X Radix-64 Conversion

Appendix Y The Base Rate Fallacy

Glossary

NOTATION

Symbol Expression Meaning

D, K D(K, Y) Symmetric decryption of ciphertext Y using secret key K

D, PRa D(PRa, Y) Asymmetric decryption of ciphertext Y using A’s private key PRa

D, PUa D(PUa, Y) Asymmetric decryption of ciphertext Y using A’s public key PUa

E, K E(K, X) Symmetric encryption of plaintext X using secret key K

E, PRa E(PRa, X) Asymmetric encryption of plaintext X using A’s private key PRa

E, PUa E(PUa, X) Asymmetric encryption of plaintext X using A’s public key PUa

K Secret key

PRa Private key of user A

PUa Public key of user A

MAC, K MAC(K, X) Message authentication code of message X using secret key K

GF(p) The finite field of order p, where p is prime.The field is defined as the set Zp together with the arithmetic operations modulo p.

GF(2n) The finite field of order 2n

Zn Set of nonnegative integers less than n

gcd gcd(i, j) Greatest common divisor; the largest positive integer that divides both i and j with no remainder on division.

mod a mod m Remainder after division of a by m

mod, K a K b (mod m) a mod m = b mod m

mod, [ a [ b (mod m) a mod m ≠ b mod m

dlog dloga,p(b) Discrete logarithm of the number b for the base a (mod p)

w f(n) The number of positive integers less than n and relatively prime to n. This is Euler’s totient function.

Σ a n

i=1 ai a1 + a2 + g + an

Π q n

i=1 ai a1 * a2 * g * an

� i � j i divides j, which means that there is no remainder when j is divided by i

� , � �a � Absolute value of a

10

Hiva-Network.Com

http://www.hiva-network.com/

NOTATION 11

Symbol Expression Meaning

} x } y x concatenated with y

≈ x ≈ y x is approximately equal to y

⊕ x⊕ y Exclusive-OR of x and y for single-bit variables; Bitwise exclusive-OR of x and y for multiple-bit variables

:, ; :x; The largest integer less than or equal to x ∈ x∈ S The element x is contained in the set S.

· A · (a1, a2, c ak)

The integer A corresponds to the sequence of integers (a1, a2, c ak)

PREFACE

WHAT’S NEW IN THE SEVENTH EDITION

In the four years since the sixth edition of this book was published, the field has seen contin- ued innovations and improvements. In this new edition, I try to capture these changes while maintaining a broad and comprehensive coverage of the entire field. To begin this process of revision, the sixth edition of this book was extensively reviewed by a number of professors who teach the subject and by professionals working in the field. The result is that, in many places, the narrative has been clarified and tightened, and illustrations have been improved.

Beyond these refinements to improve pedagogy and user-friendliness, there have been substantive changes throughout the book. Roughly the same chapter organization has been retained, but much of the material has been revised and new material has been added. The most noteworthy changes are as follows:

■ Fundamental security design principles: Chapter 1 includes a new section discussing the security design principles listed as fundamental by the National Centers of Academic Excellence in Information Assurance/Cyber Defense, which is jointly sponsored by the U.S. National Security Agency and the U.S. Department of Homeland Security.

■ Attack surfaces and attack trees: Chapter 1 includes a new section describing these two concepts, which are useful in evaluating and classifying security threats.

■ Number theory coverage: The material on number theory has been consolidated into a single chapter, Chapter 2. This makes for a convenient reference. The relevant portions of Chapter 2 can be assigned as needed.

■ Finite fields: The chapter on finite fields has been revised and expanded with addi- tional text and new figures to enhance understanding.

■ Format-preserving encryption: This relatively new mode of encryption is enjoying increasing commercial success. A new section in Chapter 7 covers this method.

■ Conditioning and health testing for true random number generators: Chapter 8 now provides coverage of these important topics.

■ User authentication model: Chapter 15 includes a new description of a general model for user authentication, which helps to unify the discussion of the various approaches to user authentication.

■ Cloud security: The material on cloud security in Chapter 16 has been updated and expanded to reflect its importance and recent developments.

■ Transport Layer Security (TLS): The treatment of TLS in Chapter 17 has been updated, reorganized to improve clarity, and now includes a discussion of the new TLS version 1.3.

■ Email Security: Chapter 19 has been completely rewritten to provide a comprehensive and up-to-date discussion of email security. It includes:

— New: discussion of email threats and a comprehensive approach to email security.

— New: discussion of STARTTLS, which provides confidentiality and authentication for SMTP.

12

PREFACE 13

— Revised: treatment of S/MIME has been updated to reflect the latest version 3.2.

— New: discussion of DNSSEC and its role in supporting email security.

— New: discussion of DNS-based Authentication of Named Entities (DANE) and the use of this approach to enhance security for certificate use in SMTP and S/MIME.

— New: discussion of Sender Policy Framework (SPF), which is the standardized way for a sending domain to identify and assert the mail senders for a given domain.

— Revised: discussion of DomainKeys Identified Mail (DKIM) has been revised.

— New: discussion of Domain-based Message Authentication, Reporting, and Confor- mance (DMARC) allows email senders to specify policy on how their mail should be handled, the types of reports that receivers can send back, and the frequency those reports should be sent.

OBJECTIVES

The subject, and therefore this book, draws on a variety of disciplines. In particular, it is impossible to appreciate the significance of some of the techniques discussed in this book without a basic understanding of number theory and some results from probability theory. Nevertheless, an attempt has been made to make the book self-contained. The book not only presents the basic mathematical results that are needed but provides the reader with an intuitive understanding of those results. Such background material is introduced as needed. This approach helps to motivate the material that is introduced, and the author considers this preferable to simply presenting all of the mathematical material in a lump at the beginning of the book.

SUPPORT OF ACM/IEEE COMPUTER SCIENCE CURRICULA 2013

The book is intended for both academic and professional audiences. As a textbook, it is intended as a one-semester undergraduate course in cryptography and network security for computer science, computer engineering, and electrical engineering majors. The changes to this edition are intended to provide support of the ACM/IEEE Computer Science Curricula 2013 (CS2013). CS2013 adds Information Assurance and Security (IAS) to the curriculum rec- ommendation as one of the Knowledge Areas in the Computer Science Body of Knowledge. The document states that IAS is now part of the curriculum recommendation because of the critical role of IAS in computer science education. CS2013 divides all course work into three categories: Core-Tier 1 (all topics should be included in the curriculum), Core-Tier-2 (all or almost all topics should be included), and elective (desirable to provide breadth and depth). In the IAS area, CS2013 recommends topics in Fundamental Concepts and Network Security

It is the purpose of this book to provide a practical survey of both the principles and practice of cryptography and network security. In the first part of the book, the basic issues to be addressed by a network security capability are explored by providing a tutorial and survey of cryptography and network security technology. The latter part of the book deals with the practice of network security: practical applications that have been implemented and are in use to provide network security.

14 PREFACE

in Tier 1 and Tier 2, and Cryptography topics as elective. This text covers virtually all of the topics listed by CS2013 in these three categories.

The book also serves as a basic reference volume and is suitable for self-study.

PLAN OF THE TEXT

The book is divided into eight parts.

■ Background

■ Symmetric Ciphers

■ Asymmetric Ciphers

■ Cryptographic Data Integrity Algorithms

■ Mutual Trust

■ Network and Internet Security

■ System Security

■ Legal and Ethical Issues

The book includes a number of pedagogic features, including the use of the computer algebra system Sage and numerous figures and tables to clarify the discussions. Each chap- ter includes a list of key words, review questions, homework problems, and suggestions for further reading. The book also includes an extensive glossary, a list of frequently used acronyms, and a bibliography. In addition, a test bank is available to instructors.

INSTRUCTOR SUPPORT MATERIALS

The major goal of this text is to make it as effective a teaching tool for this exciting and fast-moving subject as possible. This goal is reflected both in the structure of the book and in the supporting material. The text is accompanied by the following supplementary material that will aid the instructor:

■ Solutions manual: Solutions to all end-of-chapter Review Questions and Problems.

■ Projects manual: Suggested project assignments for all of the project categories listed below.

■ PowerPoint slides: A set of slides covering all chapters, suitable for use in lecturing.

■ PDF files: Reproductions of all figures and tables from the book.

■ Test bank: A chapter-by-chapter set of questions with a separate file of answers.

■ Sample syllabuses: The text contains more material than can be conveniently covered in one semester. Accordingly, instructors are provided with several sample syllabuses that guide the use of the text within limited time.

All of these support materials are available at the Instructor Resource Center (IRC) for this textbook, which can be reached through the publisher’s Web site www.pearsonglobaleditions.com/stallings. To gain access to the IRC, please contact your local Pearson sales representative.

http://www.pearsonglobaleditions.com/stallings

PREFACE 15

PROJECTS AND OTHER STUDENT EXERCISES

For many instructors, an important component of a cryptography or network security course is a project or set of projects by which the student gets hands-on experience to reinforce concepts from the text. This book provides an unparalleled degree of support, including a projects component in the course. The IRC not only includes guidance on how to assign and structure the projects, but also includes a set of project assignments that covers a broad range of topics from the text:

■ Sage projects: Described in the next section.

■ Hacking project: Exercise designed to illuminate the key issues in intrusion detection and prevention.

■ Block cipher projects: A lab that explores the operation of the AES encryption algo- rithm by tracing its execution, computing one round by hand, and then exploring the various block cipher modes of use. The lab also covers DES. In both cases, an online Java applet is used (or can be downloaded) to execute AES or DES.

■ Lab exercises: A series of projects that involve programming and experimenting with concepts from the book.

■ Research projects: A series of research assignments that instruct the student to research a particular topic on the Internet and write a report.

■ Programming projects: A series of programming projects that cover a broad range of topics and that can be implemented in any suitable language on any platform.

■ Practical security assessments: A set of exercises to examine current infrastructure and practices of an existing organization.

■ Firewall projects: A portable network firewall visualization simulator, together with exercises for teaching the fundamentals of firewalls.

■ Case studies: A set of real-world case studies, including learning objectives, case description, and a series of case discussion questions.

■ Writing assignments: A set of suggested writing assignments, organized by chapter.

■ Reading/report assignments: A list of papers in the literature—one for each chapter— that can be assigned for the student to read and then write a short report.

This diverse set of projects and other student exercises enables the instructor to use the book as one component in a rich and varied learning experience and to tailor a course plan to meet the specific needs of the instructor and students. See Appendix A in this book for details.

THE SAGE COMPUTER ALGEBRA SYSTEM

One of the most important features of this book is the use of Sage for cryptographic examples and homework assignments. Sage is an open-source, multiplatform, freeware package that implements a very powerful, flexible, and easily learned mathematics and computer algebra system. Unlike competing systems (such as Mathematica, Maple, and MATLAB), there are

16 PREFACE

no licensing agreements or fees involved. Thus, Sage can be made available on computers and networks at school, and students can individually download the software to their own personal computers for use at home. Another advantage of using Sage is that students learn a powerful, flexible tool that can be used for virtually any mathematical application, not just cryptography.

The use of Sage can make a significant difference to the teaching of the mathematics of cryptographic algorithms. This book provides a large number of examples of the use of Sage covering many cryptographic concepts in Appendix B, which is included in this book.

Appendix C lists exercises in each of these topic areas to enable the student to gain hands-on experience with cryptographic algorithms. This appendix is available to instruc- tors at the IRC for this book. Appendix C includes a section on how to download and get started with Sage, a section on programming with Sage, and exercises that can be assigned to students in the following categories:

■ Chapter 2—Number Theory and Finite Fields: Euclidean and extended Euclidean algorithms, polynomial arithmetic, GF(24), Euler’s Totient function, Miller–Rabin, fac- toring, modular exponentiation, discrete logarithm, and Chinese remainder theorem.

■ Chapter 3—Classical Encryption: Affine ciphers and the Hill cipher.

■ Chapter 4—Block Ciphers and the Data Encryption Standard: Exercises based on SDES.

■ Chapter 6—Advanced Encryption Standard: Exercises based on SAES.

■ Chapter 8—Pseudorandom Number Generation and Stream Ciphers: Blum Blum Shub, linear congruential generator, and ANSI X9.17 PRNG.

■ Chapter 9—Public-Key Cryptography and RSA: RSA encrypt/decrypt and signing.

■ Chapter 10—Other Public-Key Cryptosystems: Diffie–Hellman, elliptic curve.

■ Chapter 11—Cryptographic Hash Functions: Number-theoretic hash function.

■ Chapter 13—Digital Signatures: DSA.

ONLINE DOCUMENTS FOR STUDENTS

For this new edition, a tremendous amount of original supporting material for students has been made available online.

Purchasing this textbook new also grants the reader six months of access to the Companion Website, which includes the following materials:

■ Online chapters: To limit the size and cost of the book, four chapters of the book are provided in PDF format. This includes three chapters on computer security and one on legal and ethical issues. The chapters are listed in this book’s table of contents.

■ Online appendices: There are numerous interesting topics that support material found in the text but whose inclusion is not warranted in the printed text. A total of 20 online appendices cover these topics for the interested student. The appendices are listed in this book’s table of contents.

PREFACE 17

■ Homework problems and solutions: To aid the student in understanding the material, a separate set of homework problems with solutions are available.

■ Key papers: A number of papers from the professional literature, many hard to find, are provided for further reading.

■ Supporting documents: A variety of other useful documents are referenced in the text and provided online.

■ Sage code: The Sage code from the examples in Appendix B is useful in case the student wants to play around with the examples.

To access the Companion Website, follow the instructions for “digital resources for students” found in the front of this book.

ACKNOWLEDGMENTS

This new edition has benefited from review by a number of people who gave generously of their time and expertise. The following professors reviewed all or a large part of the manuscript: Hossein Beyzavi (Marymount University), Donald F. Costello (University of Nebraska–Lincoln), James Haralambides (Barry University), Anand Seetharam (California State University at Monterey Bay), Marius C. Silaghi (Florida Institute of Technology), Shambhu Upadhyaya (University at Buffalo), Zhengping Wu (California State University at San Bernardino), Liangliang Xiao (Frostburg State University), Seong-Moo (Sam) Yoo (The University of Alabama in Huntsville), and Hong Zhang (Armstrong State University).

Thanks also to the people who provided detailed technical reviews of one or more chapters: Dino M. Amaral, Chris Andrew, Prof. (Dr). C. Annamalai, Andrew Bain, Riccardo Bernardini, Olivier Blazy, Zervopoulou Christina, Maria Christofi, Dhananjoy Dey, Mario Emmanuel, Mike Fikuart, Alexander Fries, Pierpaolo Giacomin, Pedro R. M. Inácio, Daniela Tamy Iwassa, Krzysztof Janowski, Sergey Katsev, Adnan Kilic, Rob Knox, Mina Pourdashty, Yuri Poeluev, Pritesh Prajapati, Venkatesh Ramamoorthy, Andrea Razzini, Rami Rosen, Javier Scodelaro, Jamshid Shokrollahi, Oscar So, and David Tillemans.

In addition, I was fortunate to have reviews of individual topics by “subject-area gurus,” including Jesse Walker of Intel (Intel’s Digital Random Number Generator), Russ Housley of Vigil Security (key wrapping), Joan Daemen (AES), Edward F. Schaefer of Santa Clara University (Simplified AES), Tim Mathews, formerly of RSA Laboratories (S/MIME), Alfred Menezes of the University of Waterloo (elliptic curve cryptography), William Sutton, Editor/Publisher of The Cryptogram (classical encryption), Avi Rubin of Johns Hopkins University (number theory), Michael Markowitz of Information Security Corporation (SHA and DSS), Don Davis of IBM Internet Security Systems (Kerberos), Steve Kent of BBN Technologies (X.509), and Phil Zimmerman (PGP).

Nikhil Bhargava (IIT Delhi) developed the set of online homework problems and solutions. Dan Shumow of Microsoft and the University of Washington developed all of the Sage examples and assignments in Appendices B and C. Professor Sreekanth Malladi of Dakota State University developed the hacking exercises. Lawrie Brown of the Australian Defence Force Academy provided the AES/DES block cipher projects and the security assessment assignments.

18 PREFACE

Sanjay Rao and Ruben Torres of Purdue University developed the laboratory exercises that appear in the IRC. The following people contributed project assignments that appear in the instructor’s supplement: Henning Schulzrinne (Columbia University); Cetin Kaya Koc (Oregon State University); and David Balenson (Trusted Information Systems and George Washington University). Kim McLaughlin developed the test bank.

Finally, I thank the many people responsible for the publication of this book, all of whom did their usual excellent job. This includes the staff at Pearson, particularly my editor Tracy Johnson, program manager Carole Snyder, and production manager Bob Engelhardt. Thanks also to the marketing and sales staffs at Pearson, without whose efforts this book would not be in front of you.

ACKNOWLEDGMENTS FOR THE GLOBAL EDITION

Pearson would like to thank and acknowledge Somitra Kumar Sanadhya (Indraprastha Institute of Information Technology Delhi), and Somanath Tripathy (Indian Institute of Technology Patna) for contributing to the Global Edition, and Anwitaman Datta (Nanyang Technological University Singapore), Atul Kahate (Pune University), Goutam Paul (Indian Statistical Institute Kolkata), and Khyat Sharma for reviewing the Global Edition.

ABOUT THE AUTHOR

Dr. William Stallings has authored 18 titles, and counting revised editions, over 40 books on computer security, computer networking, and computer architecture. His writings have appeared in numerous publications, including the Proceedings of the IEEE, ACM Computing Reviews, and Cryptologia.

He has 13 times received the award for the best Computer Science textbook of the year from the Text and Academic Authors Association.

In over 30 years in the field, he has been a technical contributor, technical manager, and an executive with several high-technology firms. He has designed and implemented both TCP/IP-based and OSI-based protocol suites on a variety of computers and operating systems, ranging from microcomputers to mainframes. As a consultant, he has advised gov- ernment agencies, computer and software vendors, and major users on the design, selection, and use of networking software and products.

He created and maintains the Computer Science Student Resource Site at ComputerScienceStudent.com. This site provides documents and links on a variety of subjects of general interest to computer science students (and professionals). He is a member of the editorial board of Cryptologia, a scholarly journal devoted to all aspects of cryptology.

Dr. Stallings holds a PhD from MIT in computer science and a BS from Notre Dame in electrical engineering.

19

PART ONE: BACKGROUND

CHAPTER

Computer and Network Security Concepts

1.1 Computer Security Concepts

A Definition of Computer Security Examples The Challenges of Computer Security

1.2 The OSI Security Architecture

1.3 Security Attacks

Passive Attacks Active Attacks

1.4 Security Services

Authentication Access Control Data Confidentiality Data Integrity Nonrepudiation Availability Service

1.5 Security Mechanisms

1.6 Fundamental Security Design Principles

1.7 Attack Surfaces and Attack Trees

Attack Surfaces Attack Trees

1.8 A Model for Network Security

1.9 Standards

1.10 Key Terms, Review Questions, and Problems

19

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20 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS

This book focuses on two broad areas: cryptographic algorithms and protocols, which have a broad range of applications; and network and Internet security, which rely heavily on cryptographic techniques.

Cryptographic algorithms and protocols can be grouped into four main areas:

■ Symmetric encryption: Used to conceal the contents of blocks or streams of data of any size, including messages, files, encryption keys, and passwords.

■ Asymmetric encryption: Used to conceal small blocks of data, such as encryp- tion keys and hash function values, which are used in digital signatures.

■ Data integrity algorithms: Used to protect blocks of data, such as messages, from alteration.

■ Authentication protocols: These are schemes based on the use of crypto- graphic algorithms designed to authenticate the identity of entities.

The field of network and Internet security consists of measures to deter, prevent, detect, and correct security violations that involve the transmission of information. That is a broad statement that covers a host of possibilities. To give you a feel for the areas covered in this book, consider the following examples of security violations:

1. User A transmits a file to user B. The file contains sensitive information (e.g., payroll records) that is to be protected from disclosure. User C, who is not authorized to read the file, is able to monitor the transmission and capture a copy of the file during its transmission.

2. A network manager, D, transmits a message to a computer, E, under its man- agement. The message instructs computer E to update an authorization file to include the identities of a number of new users who are to be given access to that computer. User F intercepts the message, alters its contents to add or delete entries, and then forwards the message to computer E, which accepts the mes- sage as coming from manager D and updates its authorization file accordingly.

LEARNING OBJECTIVES

After studying this chapter, you should be able to:

◆ Describe the key security requirements of confidentiality, integrity, and availability.

◆ Describe the X.800 security architecture for OSI.

◆ Discuss the types of security threats and attacks that must be dealt with and give examples of the types of threats and attacks that apply to differ- ent categories of computer and network assets.

◆ Explain the fundamental security design principles.

◆ Discuss the use of attack surfaces and attack trees.

◆ List and briefly describe key organizations involved in cryptography standards.

1.1 / COMPUTER SECURITY CONCEPTS 21

3. Rather than intercept a message, user F constructs its own message with the desired entries and transmits that message to computer E as if it had come from manager D. Computer E accepts the message as coming from manager D and updates its authorization file accordingly.

4. An employee is fired without warning. The personnel manager sends a mes- sage to a server system to invalidate the employee’s account. When the invali- dation is accomplished, the server is to post a notice to the employee’s file as confirmation of the action. The employee is able to intercept the message and delay it long enough to make a final access to the server to retrieve sensitive information. The message is then forwarded, the action taken, and the confir- mation posted. The employee’s action may go unnoticed for some consider- able time.

5. A message is sent from a customer to a stockbroker with instructions for vari- ous transactions. Subsequently, the investments lose value and the customer denies sending the message.

Although this list by no means exhausts the possible types of network security viola- tions, it illustrates the range of concerns of network security.

1.1 COMPUTER SECURITY CONCEPTS

A Definition of Computer Security

The NIST Computer Security Handbook [NIST95] defines the term computer secu- rity as follows:

Computer Security: The protection afforded to an automated information system in order to attain the applicable objectives of preserving the integrity, availability, and confidentiality of information system resources (includes hardware, software, firmware, information/data, and telecommunications).

This definition introduces three key objectives that are at the heart of com- puter security:

■ Confidentiality: This term covers two related concepts:

Data1 confidentiality: Assures that private or confidential information is not made available or disclosed to unauthorized individuals.

Privacy: Assures that individuals control or influence what information re- lated to them may be collected and stored and by whom and to whom that information may be disclosed.

1RFC 4949 defines information as “facts and ideas, which can be represented (encoded) as various forms of data,” and data as “information in a specific physical representation, usually a sequence of symbols that have meaning; especially a representation of information that can be processed or produced by a computer.” Security literature typically does not make much of a distinction, nor does this book.

22 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS

■ Integrity: This term covers two related concepts:

Data integrity: Assures that information (both stored and in transmit- ted packets) and programs are changed only in a specified and authorized manner.

System integrity: Assures that a system performs its intended function in an unimpaired manner, free from deliberate or inadvertent unauthorized manipulation of the system.

■ Availability: Assures that systems work promptly and service is not denied to authorized users.

These three concepts form what is often referred to as the CIA triad. The three concepts embody the fundamental security objectives for both data and for information and computing services. For example, the NIST standard FIPS 199 (Standards for Security Categorization of Federal Information and Information Systems) lists confidentiality, integrity, and availability as the three security objec- tives for information and for information systems. FIPS 199 provides a useful char- acterization of these three objectives in terms of requirements and the definition of a loss of security in each category:

■ Confidentiality: Preserving authorized restrictions on information access and disclosure, including means for protecting personal privacy and propri- etary information. A loss of confidentiality is the unauthorized disclosure of information.

■ Integrity: Guarding against improper information modification or destruc- tion, including ensuring information nonrepudiation and authenticity. A loss of integrity is the unauthorized modification or destruction of information.

■ Availability: Ensuring timely and reliable access to and use of information. A loss of availability is the disruption of access to or use of information or an information system.

Although the use of the CIA triad to define security objectives is well estab- lished, some in the security field feel that additional concepts are needed to present a complete picture (Figure 1.1). Two of the most commonly mentioned are as follows:

Figure 1.1 Essential Network and Computer Security Requirements

Data and

services

Availability

Integrity

A ccountability

A ut

he nt

ic ity

Co nfid

ent iali

ty

1.1 / COMPUTER SECURITY CONCEPTS 23

■ Authenticity: The property of being genuine and being able to be verified and trusted; confidence in the validity of a transmission, a message, or message originator. This means verifying that users are who they say they are and that each input arriving at the system came from a trusted source.

■ Accountability: The security goal that generates the requirement for actions of an entity to be traced uniquely to that entity. This supports nonrepudia- tion, deterrence, fault isolation, intrusion detection and prevention, and after- action recovery and legal action. Because truly secure systems are not yet an achievable goal, we must be able to trace a security breach to a responsible party. Systems must keep records of their activities to permit later forensic analysis to trace security breaches or to aid in transaction disputes.

Examples

We now provide some examples of applications that illustrate the requirements just enumerated.2 For these examples, we use three levels of impact on organizations or individuals should there be a breach of security (i.e., a loss of confidentiality, integ- rity, or availability). These levels are defined in FIPS PUB 199:

■ Low: The loss could be expected to have a limited adverse effect on organi- zational operations, organizational assets, or individuals. A limited adverse effect means that, for example, the loss of confidentiality, integrity, or avail- ability might (i) cause a degradation in mission capability to an extent and duration that the organization is able to perform its primary functions, but the effectiveness of the functions is noticeably reduced; (ii) result in minor dam- age to organizational assets; (iii) result in minor financial loss; or (iv) result in minor harm to individuals.

■ Moderate: The loss could be expected to have a serious adverse effect on organizational operations, organizational assets, or individuals. A serious adverse effect means that, for example, the loss might (i) cause a signifi- cant degradation in mission capability to an extent and duration that the organization is able to perform its primary functions, but the effectiveness of the functions is significantly reduced; (ii) result in significant damage to organizational assets; (iii) result in significant financial loss; or (iv) result in significant harm to individuals that does not involve loss of life or serious, life-threatening injuries.

■ High: The loss could be expected to have a severe or catastrophic adverse effect on organizational operations, organizational assets, or individuals. A severe or catastrophic adverse effect means that, for example, the loss might (i) cause a severe degradation in or loss of mission capability to an extent and duration that the organization is not able to perform one or more of its primary functions; (ii) result in major damage to organizational assets; (iii) result in major financial loss; or (iv) result in severe or catastrophic harm to individuals involving loss of life or serious, life-threatening injuries.

2These examples are taken from a security policy document published by the Information Technology Security and Privacy Office at Purdue University.

24 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS

CONFIDENTIALITY Student grade information is an asset whose confidentiality is considered to be highly important by students. In the United States, the release of such information is regulated by the Family Educational Rights and Privacy Act (FERPA). Grade information should only be available to students, their parents, and employees that require the information to do their job. Student enrollment information may have a moderate confidentiality rating. While still covered by FERPA, this information is seen by more people on a daily basis, is less likely to be targeted than grade information, and results in less damage if disclosed. Directory information, such as lists of students or faculty or departmental lists, may be as- signed a low confidentiality rating or indeed no rating. This information is typically freely available to the public and published on a school’s Web site.

INTEGRITY Several aspects of integrity are illustrated by the example of a hospital patient’s allergy information stored in a database. The doctor should be able to trust that the information is correct and current. Now suppose that an employee (e.g., a nurse) who is authorized to view and update this information deliberately falsifies the data to cause harm to the hospital. The database needs to be restored to a trusted basis quickly, and it should be possible to trace the error back to the person responsible. Patient allergy information is an example of an asset with a high requirement for integrity. Inaccurate information could result in serious harm or death to a patient and expose the hospital to massive liability.

An example of an asset that may be assigned a moderate level of integrity requirement is a Web site that offers a forum to registered users to discuss some specific topic. Either a registered user or a hacker could falsify some entries or deface the Web site. If the forum exists only for the enjoyment of the users, brings in little or no advertising revenue, and is not used for something important such as research, then potential damage is not severe. The Web master may experience some data, financial, and time loss.

An example of a low integrity requirement is an anonymous online poll. Many Web sites, such as news organizations, offer these polls to their users with very few safeguards. However, the inaccuracy and unscientific nature of such polls is well understood.

AVAILABILITY The more critical a component or service, the higher is the level of availability required. Consider a system that provides authentication services for critical systems, applications, and devices. An interruption of service results in the inability for customers to access computing resources and staff to access the re- sources they need to perform critical tasks. The loss of the service translates into a large financial loss in lost employee productivity and potential customer loss.

An example of an asset that would typically be rated as having a moderate availability requirement is a public Web site for a university; the Web site provides information for current and prospective students and donors. Such a site is not a critical component of the university’s information system, but its unavailability will cause some embarrassment.

An online telephone directory lookup application would be classified as a low availability requirement. Although the temporary loss of the application may be an annoyance, there are other ways to access the information, such as a hardcopy directory or the operator.

1.1 / COMPUTER SECURITY CONCEPTS 25

The Challenges of Computer Security

Computer and network security is both fascinating and complex. Some of the reasons follow:

1. Security is not as simple as it might first appear to the novice. The require- ments seem to be straightforward; indeed, most of the major requirements for security services can be given self-explanatory, one-word labels: confidential- ity, authentication, nonrepudiation, or integrity. But the mechanisms used to meet those requirements can be quite complex, and understanding them may involve rather subtle reasoning.

2. In developing a particular security mechanism or algorithm, one must always consider potential attacks on those security features. In many cases, successful attacks are designed by looking at the problem in a completely different way, therefore exploiting an unexpected weakness in the mechanism.

3. Because of point 2, the procedures used to provide particular services are often counterintuitive. Typically, a security mechanism is complex, and it is not obvious from the statement of a particular requirement that such elaborate measures are needed. It is only when the various aspects of the threat are con- sidered that elaborate security mechanisms make sense.

4. Having designed various security mechanisms, it is necessary to decide where to use them. This is true both in terms of physical placement (e.g., at what points in a network are certain security mechanisms needed) and in a logical sense (e.g., at what layer or layers of an architecture such as TCP/IP [Transmission Control Protocol/Internet Protocol] should mechanisms be placed).

5. Security mechanisms typically involve more than a particular algorithm or protocol. They also require that participants be in possession of some secret in- formation (e.g., an encryption key), which raises questions about the creation, distribution, and protection of that secret information. There also may be a re- liance on communications protocols whose behavior may complicate the task of developing the security mechanism. For example, if the proper functioning of the security mechanism requires setting time limits on the transit time of a message from sender to receiver, then any protocol or network that introduces variable, unpredictable delays may render such time limits meaningless.

6. Computer and network security is essentially a battle of wits between a per- petrator who tries to find holes and the designer or administrator who tries to close them. The great advantage that the attacker has is that he or she need only find a single weakness, while the designer must find and eliminate all weaknesses to achieve perfect security.

7. There is a natural tendency on the part of users and system managers to per- ceive little benefit from security investment until a security failure occurs.

8. Security requires regular, even constant, monitoring, and this is difficult in today’s short-term, overloaded environment.

9. Security is still too often an afterthought to be incorporated into a system after the design is complete rather than being an integral part of the design process.

26 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS

10. Many users and even security administrators view strong security as an impediment to efficient and user-friendly operation of an information system or use of information.

The difficulties just enumerated will be encountered in numerous ways as we examine the various security threats and mechanisms throughout this book.

1.2 THE OSI SECURITY ARCHITECTURE

To assess effectively the security needs of an organization and to evaluate and choose various security products and policies, the manager responsible for security needs some systematic way of defining the requirements for security and character- izing the approaches to satisfying those requirements. This is difficult enough in a centralized data processing environment; with the use of local and wide area net- works, the problems are compounded.

ITU-T3 Recommendation X.800, Security Architecture for OSI, defines such a systematic approach.4 The OSI security architecture is useful to managers as a way of organizing the task of providing security. Furthermore, because this architecture was developed as an international standard, computer and communications vendors have developed security features for their products and services that relate to this structured definition of services and mechanisms.

For our purposes, the OSI security architecture provides a useful, if abstract, overview of many of the concepts that this book deals with. The OSI security archi- tecture focuses on security attacks, mechanisms, and services. These can be defined briefly as

■ Security attack: Any action that compromises the security of information owned by an organization.

■ Security mechanism: A process (or a device incorporating such a process) that is designed to detect, prevent, or recover from a security attack.

■ Security service: A processing or communication service that enhances the security of the data processing systems and the information transfers of an organization. The services are intended to counter security attacks, and they make use of one or more security mechanisms to provide the service.

In the literature, the terms threat and attack are commonly used to mean more or less the same thing. Table 1.1 provides definitions taken from RFC 4949, Internet Security Glossary.

3The International Telecommunication Union (ITU) Telecommunication Standardization Sector (ITU-T) is a United Nations-sponsored agency that develops standards, called Recommendations, relating to tele- communications and to open systems interconnection (OSI). 4The OSI security architecture was developed in the context of the OSI protocol architecture, which is described in Appendix L. However, for our purposes in this chapter, an understanding of the OSI proto- col architecture is not required.

1.3 / SECURITY ATTACKS 27

1.3 SECURITY ATTACKS

A useful means of classifying security attacks, used both in X.800 and RFC 4949, is in terms of passive attacks and active attacks (Figure 1.2). A passive attack attempts to learn or make use of information from the system but does not affect system re- sources. An active attack attempts to alter system resources or affect their operation.

Passive Attacks

Passive attacks (Figure 1.2a) are in the nature of eavesdropping on, or monitoring of, transmissions. The goal of the opponent is to obtain information that is being transmitted. Two types of passive attacks are the release of message contents and traffic analysis.

The release of message contents is easily understood. A telephone conver- sation, an electronic mail message, and a transferred file may contain sensitive or confidential information. We would like to prevent an opponent from learning the contents of these transmissions.

A second type of passive attack, traffic analysis, is subtler. Suppose that we had a way of masking the contents of messages or other information traffic so that opponents, even if they captured the message, could not extract the information from the message. The common technique for masking contents is encryption. If we had encryption protection in place, an opponent might still be able to observe the pattern of these messages. The opponent could determine the location and identity of communicating hosts and could observe the frequency and length of messages being exchanged. This information might be useful in guessing the nature of the communication that was taking place.

Passive attacks are very difficult to detect, because they do not involve any alteration of the data. Typically, the message traffic is sent and received in an appar- ently normal fashion, and neither the sender nor receiver is aware that a third party has read the messages or observed the traffic pattern. However, it is feasible to pre- vent the success of these attacks, usually by means of encryption. Thus, the empha- sis in dealing with passive attacks is on prevention rather than detection.

Active Attacks

Active attacks (Figure 1.2b) involve some modification of the data stream or the creation of a false stream and can be subdivided into four categories: masquerade, replay, modification of messages, and denial of service.

Threat A potential for violation of security, which exists when there is a circumstance, capability, action, or event that could breach security and cause harm. That is, a threat is a possible danger that might exploit a vulnerability.

Attack An assault on system security that derives from an intelligent threat; that is, an intelligent act that is a deliberate attempt (especially in the sense of a method or technique) to evade security services and violate the security policy of a system.

Table 1.1 Threats and Attacks (RFC 4949)

28 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS

A masquerade takes place when one entity pretends to be a different entity (path 2 of Figure 1.2b is active). A masquerade attack usually includes one of the other forms of active attack. For example, authentication sequences can be captured and replayed after a valid authentication sequence has taken place, thus enabling an authorized entity with few privileges to obtain extra privileges by impersonating an entity that has those privileges.

Replay involves the passive capture of a data unit and its subsequent retrans- mission to produce an unauthorized effect (paths 1, 2, and 3 active).

Modification of messages simply means that some portion of a legitimate mes- sage is altered, or that messages are delayed or reordered, to produce an unauthor- ized effect (paths 1 and 2 active). For example, a message meaning “Allow John Smith to read confidential file accounts” is modified to mean “Allow Fred Brown to read confidential file accounts.”

Figure 1.2 Security Attacks

(a) Passive attacks

Alice

(b) Active attacks

Bob

Darth

Bob

Darth

Alice

Internet or other communications facility

Internet or other communications facility

1 2 3

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1.4 / SECURITY SERVICES 29

The denial of service prevents or inhibits the normal use or management of communications facilities (path 3 active). This attack may have a specific target; for example, an entity may suppress all messages directed to a particular destination (e.g., the security audit service). Another form of service denial is the disruption of an entire network, either by disabling the network or by overloading it with mes- sages so as to degrade performance.

Active attacks present the opposite characteristics of passive attacks. Whereas passive attacks are difficult to detect, measures are available to prevent their success. On the other hand, it is quite difficult to prevent active attacks absolutely because of the wide variety of potential physical, software, and network vulnerabilities. Instead, the goal is to detect active attacks and to recover from any disruption or delays caused by them. If the detection has a deterrent effect, it may also contribute to prevention.

1.4 SECURITY SERVICES

X.800 defines a security service as a service that is provided by a protocol layer of communicating open systems and that ensures adequate security of the systems or of data transfers. Perhaps a clearer definition is found in RFC 4949, which provides the following definition: a processing or communication service that is provided by a system to give a specific kind of protection to system resources; security services implement security policies and are implemented by security mechanisms.

X.800 divides these services into five categories and fourteen specific services (Table 1.2). We look at each category in turn.5

Authentication

The authentication service is concerned with assuring that a communication is au- thentic. In the case of a single message, such as a warning or alarm signal, the function of the authentication service is to assure the recipient that the message is from the source that it claims to be from. In the case of an ongoing interaction, such as the con- nection of a terminal to a host, two aspects are involved. First, at the time of connec- tion initiation, the service assures that the two entities are authentic, that is, that each is the entity that it claims to be. Second, the service must assure that the connection is not interfered with in such a way that a third party can masquerade as one of the two legitimate parties for the purposes of unauthorized transmission or reception.

Two specific authentication services are defined in X.800:

■ Peer entity authentication: Provides for the corroboration of the identity of a peer entity in an association. Two entities are considered peers if they imple- ment to same protocol in different systems; for example two TCP modules in two communicating systems. Peer entity authentication is provided for

5There is no universal agreement about many of the terms used in the security literature. For example, the term integrity is sometimes used to refer to all aspects of information security. The term authentication is sometimes used to refer both to verification of identity and to the various functions listed under integrity in this chapter. Our usage here agrees with both X.800 and RFC 4949.

30 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS

AUTHENTICATION

The assurance that the communicating entity is the one that it claims to be.

Peer Entity Authentication Used in association with a logical connection to provide confidence in the identity of the entities connected.

Data-Origin Authentication In a connectionless transfer, provides assurance that the source of received data is as claimed.

ACCESS CONTROL

The prevention of unauthorized use of a resource (i.e., this service controls who can have access to a resource, under what conditions access can occur, and what those accessing the resource are allowed to do).

DATA CONFIDENTIALITY

The protection of data from unauthorized disclosure.

Connection Confidentiality The protection of all user data on a connection.

Connectionless Confidentiality The protection of all user data in a single data block.

Selective-Field Confidentiality The confidentiality of selected fields within the user data on a connection or in a single data block.

Traffic-Flow Confidentiality The protection of the information that might be derived from observation of traffic flows.

DATA INTEGRITY

The assurance that data received are exactly as sent by an authorized entity (i.e., contain no modi- fication, insertion, deletion, or replay).

Connection Integrity with Recovery Provides for the integrity of all user data on a connec- tion and detects any modification, insertion, deletion, or replay of any data within an entire data sequence, with recovery attempted.

Connection Integrity without Recovery As above, but provides only detection without recovery.

Selective-Field Connection Integrity Provides for the integrity of selected fields within the user data of a data block transferred over a connec- tion and takes the form of determination of whether the selected fields have been modified, inserted, deleted, or replayed.

Connectionless Integrity Provides for the integrity of a single connectionless data block and may take the form of detection of data modification. Additionally, a limited form of replay detection may be provided.

Selective-Field Connectionless Integrity Provides for the integrity of selected fields within a single connectionless data block; takes the form of determination of whether the selected fields have been modified.

NONREPUDIATION

Provides protection against denial by one of the entities involved in a communication of having par- ticipated in all or part of the communication.

Nonrepudiation, Origin Proof that the message was sent by the specified party.

Nonrepudiation, Destination Proof that the message was received by the specified party.

Table 1.2 Security Services (X.800)

use at the establishment of, or at times during the data transfer phase of, a connection. It attempts to provide confidence that an entity is not performing either a masquerade or an unauthorized replay of a previous connection.

■ Data origin authentication: Provides for the corroboration of the source of a data unit. It does not provide protection against the duplication or modifica- tion of data units. This type of service supports applications like electronic mail, where there are no prior interactions between the communicating entities.

1.4 / SECURITY SERVICES 31

Access Control

In the context of network security, access control is the ability to limit and control the access to host systems and applications via communications links. To achieve this, each entity trying to gain access must first be identified, or authenticated, so that access rights can be tailored to the individual.

Data Confidentiality

Confidentiality is the protection of transmitted data from passive attacks. With re- spect to the content of a data transmission, several levels of protection can be iden- tified. The broadest service protects all user data transmitted between two users over a period of time. For example, when a TCP connection is set up between two systems, this broad protection prevents the release of any user data transmitted over the TCP connection. Narrower forms of this service can also be defined, including the protection of a single message or even specific fields within a message. These refinements are less useful than the broad approach and may even be more complex and expensive to implement.

The other aspect of confidentiality is the protection of traffic flow from analysis. This requires that an attacker not be able to observe the source and desti- nation, frequency, length, or other characteristics of the traffic on a communications facility.

Data Integrity

As with confidentiality, integrity can apply to a stream of messages, a single mes- sage, or selected fields within a message. Again, the most useful and straightforward approach is total stream protection.

A connection-oriented integrity service, one that deals with a stream of mes- sages, assures that messages are received as sent with no duplication, insertion, modification, reordering, or replays. The destruction of data is also covered under this service. Thus, the connection-oriented integrity service addresses both mes- sage stream modification and denial of service. On the other hand, a connection- less integrity service, one that deals with individual messages without regard to any larger context, generally provides protection against message modification only.

We can make a distinction between service with and without recovery. Because the integrity service relates to active attacks, we are concerned with detection rather than prevention. If a violation of integrity is detected, then the service may simply report this violation, and some other portion of software or human intervention is required to recover from the violation. Alternatively, there are mechanisms avail- able to recover from the loss of integrity of data, as we will review subsequently. The incorporation of automated recovery mechanisms is, in general, the more attractive alternative.

Nonrepudiation

Nonrepudiation prevents either sender or receiver from denying a transmitted mes- sage. Thus, when a message is sent, the receiver can prove that the alleged sender in fact sent the message. Similarly, when a message is received, the sender can prove that the alleged receiver in fact received the message.

32 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS

Availability Service

Both X.800 and RFC 4949 define availability to be the property of a system or a system resource being accessible and usable upon demand by an authorized system entity, according to performance specifications for the system (i.e., a system is avail- able if it provides services according to the system design whenever users request them). A variety of attacks can result in the loss of or reduction in availability. Some of these attacks are amenable to automated countermeasures, such as authentica- tion and encryption, whereas others require some sort of physical action to prevent or recover from loss of availability of elements of a distributed system.

X.800 treats availability as a property to be associated with various security services. However, it makes sense to call out specifically an availability service. An availability service is one that protects a system to ensure its availability. This ser- vice addresses the security concerns raised by denial-of-service attacks. It depends on proper management and control of system resources and thus depends on access control service and other security services.

1.5 SECURITY MECHANISMS

Table 1.3 lists the security mechanisms defined in X.800. The mechanisms are divided into those that are implemented in a specific protocol layer, such as TCP or an application-layer protocol, and those that are not specific to any particular pro- tocol layer or security service. These mechanisms will be covered in the appropriate

SPECIFIC SECURITY MECHANISMS May be incorporated into the appropriate protocol layer in order to provide some of the OSI security services.

Encipherment The use of mathematical algorithms to transform data into a form that is not readily intelligible. The transformation and subsequent recovery of the data depend on an algorithm and zero or more encryption keys.

Digital Signature Data appended to, or a cryptographic transformation of, a data unit that allows a recipient of the data unit to prove the source and integrity of the data unit and protect against forgery (e.g., by the recipient).

Access Control A variety of mechanisms that enforce access rights to resources.

Data Integrity A variety of mechanisms used to assure the integrity of a data unit or stream of data units.

PERVASIVE SECURITY MECHANISMS

Mechanisms that are not specific to any particular OSI security service or protocol layer.

Trusted Functionality That which is perceived to be correct with respect to some criteria (e.g., as established by a security policy).

Security Label The marking bound to a resource (which may be a data unit) that names or designates the security attri- butes of that resource.

Event Detection Detection of security-relevant events.

Security Audit Trail Data collected and potentially used to facilitate a security audit, which is an independent review and examination of system records and activities.

Security Recovery Deals with requests from mechanisms, such as event handling and management functions, and takes recovery actions.

Table 1.3 Security Mechanisms (X.800)

1.5 / SECURITY MECHANISMS 33

places in the book. So we do not elaborate now, except to comment on the defini- tion of encipherment. X.800 distinguishes between reversible encipherment mech- anisms and irreversible encipherment mechanisms. A reversible encipherment mechanism is simply an encryption algorithm that allows data to be encrypted and subsequently decrypted. Irreversible encipherment mechanisms include hash algo- rithms and message authentication codes, which are used in digital signature and message authentication applications.

Table 1.4, based on one in X.800, indicates the relationship between security services and security mechanisms.

SPECIFIC SECURITY MECHANISMS

Authentication Exchange A mechanism intended to ensure the identity of an entity by means of information exchange.

Traffic Padding The insertion of bits into gaps in a data stream to frustrate traffic analysis attempts.

Routing Control Enables selection of particular physically secure routes for certain data and allows routing changes, especially when a breach of security is suspected.

Notarization The use of a trusted third party to assure certain properties of a data exchange.

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Peer entity authentication

SERVICE

MECHANISM

En cip

he rm

en t

Di git

al sig

na tur

e

Ac ces

s c on

tro l

Da ta

int eg

rit y

Au the

nti cat

ion ex

ch an

ge

Tr affi

c p ad

din g

Ro uti

ng co

ntr ol

No tar

iza tio

n

Data origin authentication

Access control

Confidentiality

Traffic flow confidentiality

Data integrity

Nonrepudiation

Availability

Table 1.4 Relationship Between Security Services and Mechanisms

34 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS

1.6 FUNDAMENTAL SECURITY DESIGN PRINCIPLES

Despite years of research and development, it has not been possible to develop security design and implementation techniques that systematically exclude security flaws and prevent all unauthorized actions. In the absence of such foolproof tech- niques, it is useful to have a set of widely agreed design principles that can guide the development of protection mechanisms. The National Centers of Academic Excellence in Information Assurance/Cyber Defense, which is jointly sponsored by the U.S. National Security Agency and the U.S. Department of Homeland Security, list the following as fundamental security design principles [NCAE13]:

■ Economy of mechanism

■ Fail-safe defaults

■ Complete mediation

■ Open design

■ Separation of privilege

■ Least privilege

■ Least common mechanism

■ Psychological acceptability

■ Isolation

■ Encapsulation

■ Modularity

■ Layering

■ Least astonishment

The first eight listed principles were first proposed in [SALT75] and have withstood the test of time. In this section, we briefly discuss each principle.

Economy of mechanism means that the design of security measures embod- ied in both hardware and software should be as simple and small as possible. The motivation for this principle is that relatively simple, small design is eas- ier to test and verify thoroughly. With a complex design, there are many more opportunities for an adversary to discover subtle weaknesses to exploit that may be difficult to spot ahead of time. The more complex the mechanism, the more likely it is to possess exploitable flaws. Simple mechanisms tend to have fewer exploitable flaws and require less maintenance. Further, because configuration management issues are simplified, updating or replacing a simple mechanism becomes a less intensive process. In practice, this is perhaps the most difficult principle to honor. There is a constant demand for new features in both hard- ware and software, complicating the security design task. The best that can be done is to keep this principle in mind during system design to try to eliminate unnecessary complexity.

Fail-safe defaults means that access decisions should be based on permission rather than exclusion. That is, the default situation is lack of access, and the protec- tion scheme identifies conditions under which access is permitted. This approach

1.6 / FUNDAMENTAL SECURITY DESIGN PRINCIPLES 35

exhibits a better failure mode than the alternative approach, where the default is to permit access. A design or implementation mistake in a mechanism that gives explicit permission tends to fail by refusing permission, a safe situation that can be quickly detected. On the other hand, a design or implementation mistake in a mechanism that explicitly excludes access tends to fail by allowing access, a failure that may long go unnoticed in normal use. Most file access systems and virtually all protected services on client/server systems use fail-safe defaults.

Complete mediation means that every access must be checked against the access control mechanism. Systems should not rely on access decisions retrieved from a cache. In a system designed to operate continuously, this principle requires that, if access decisions are remembered for future use, careful consideration be given to how changes in authority are propagated into such local memories. File access systems appear to provide an example of a system that complies with this principle. However, typically, once a user has opened a file, no check is made to see if permissions change. To fully implement complete mediation, every time a user reads a field or record in a file, or a data item in a database, the system must exercise access control. This resource-intensive approach is rarely used.

Open design means that the design of a security mechanism should be open rather than secret. For example, although encryption keys must be secret, encryption algorithms should be open to public scrutiny. The algorithms can then be reviewed by many experts, and users can therefore have high confidence in them. This is the philosophy behind the National Institute of Standards and Technology (NIST) program of standardizing encryption and hash algorithms, and has led to the wide- spread adoption of NIST-approved algorithms.

Separation of privilege is defined in [SALT75] as a practice in which mul- tiple privilege attributes are required to achieve access to a restricted resource. A good example of this is multifactor user authentication, which requires the use of multiple techniques, such as a password and a smart card, to authorize a user. The term is also now applied to any technique in which a program is divided into parts that are limited to the specific privileges they require in order to perform a specific task. This is used to mitigate the potential damage of a computer security attack. One example of this latter interpretation of the principle is removing high privilege operations to another process and running that process with the higher privileges required to perform its tasks. Day-to-day interfaces are executed in a lower privi- leged process.

Least privilege means that every process and every user of the system should operate using the least set of privileges necessary to perform the task. A good example of the use of this principle is role-based access control. The system security policy can identify and define the various roles of users or processes. Each role is assigned only those permissions needed to perform its functions. Each permission specifies a permitted access to a particular resource (such as read and write access to a specified file or directory, connect access to a given host and port). Unless a permission is granted explicitly, the user or process should not be able to access the protected resource. More generally, any access control system should allow each user only the privileges that are authorized for that user. There is also a temporal aspect to the least privilege principle. For example, system programs or administra- tors who have special privileges should have those privileges only when necessary;

36 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS

when they are doing ordinary activities the privileges should be withdrawn. Leaving them in place just opens the door to accidents.

Least common mechanism means that the design should minimize the func- tions shared by different users, providing mutual security. This principle helps reduce the number of unintended communication paths and reduces the amount of hardware and software on which all users depend, thus making it easier to verify if there are any undesirable security implications.

Psychological acceptability implies that the security mechanisms should not interfere unduly with the work of users, while at the same time meeting the needs of those who authorize access. If security mechanisms hinder the usability or accessibil- ity of resources, then users may opt to turn off those mechanisms. Where possible, security mechanisms should be transparent to the users of the system or at most introduce minimal obstruction. In addition to not being intrusive or burdensome, security procedures must reflect the user’s mental model of protection. If the protec- tion procedures do not make sense to the user or if the user must translate his image of protection into a substantially different protocol, the user is likely to make errors.

Isolation is a principle that applies in three contexts. First, public access sys- tems should be isolated from critical resources (data, processes, etc.) to prevent dis- closure or tampering. In cases where the sensitivity or criticality of the information is high, organizations may want to limit the number of systems on which that data is stored and isolate them, either physically or logically. Physical isolation may include ensuring that no physical connection exists between an organization’s public access information resources and an organization’s critical information. When implement- ing logical isolation solutions, layers of security services and mechanisms should be established between public systems and secure systems responsible for protecting critical resources. Second, the processes and files of individual users should be iso- lated from one another except where it is explicitly desired. All modern operating systems provide facilities for such isolation, so that individual users have separate, isolated process space, memory space, and file space, with protections for prevent- ing unauthorized access. And finally, security mechanisms should be isolated in the sense of preventing access to those mechanisms. For example, logical access control may provide a means of isolating cryptographic software from other parts of the host system and for protecting cryptographic software from tampering and the keys from replacement or disclosure.

Encapsulation can be viewed as a specific form of isolation based on object- oriented functionality. Protection is provided by encapsulating a collection of pro- cedures and data objects in a domain of its own so that the internal structure of a data object is accessible only to the procedures of the protected subsystem, and the procedures may be called only at designated domain entry points.

Modularity in the context of security refers both to the development of security functions as separate, protected modules and to the use of a modular architecture for mechanism design and implementation. With respect to the use of separate security modules, the design goal here is to provide common security functions and services, such as cryptographic functions, as common modules. For example, numerous proto- cols and applications make use of cryptographic functions. Rather than implement- ing such functions in each protocol or application, a more secure design is provided by developing a common cryptographic module that can be invoked by numerous

1.7 / ATTACK SURFACES AND ATTACK TREES 37

protocols and applications. The design and implementation effort can then focus on the secure design and implementation of a single cryptographic module and includ- ing mechanisms to protect the module from tampering. With respect to the use of a modular architecture, each security mechanism should be able to support migration to new technology or upgrade of new features without requiring an entire system redesign. The security design should be modular so that individual parts of the secu- rity design can be upgraded without the requirement to modify the entire system.

Layering refers to the use of multiple, overlapping protection approaches addressing the people, technology, and operational aspects of information systems. By using multiple, overlapping protection approaches, the failure or circumven- tion of any individual protection approach will not leave the system unprotected. We will see throughout this book that a layering approach is often used to provide multiple barriers between an adversary and protected information or services. This technique is often referred to as defense in depth.

Least astonishment means that a program or user interface should always respond in the way that is least likely to astonish the user. For example, the mechanism for authorization should be transparent enough to a user that the user has a good intui- tive understanding of how the security goals map to the provided security mechanism.

1.7 ATTACK SURFACES AND ATTACK TREES

In Section 1.3, we provided an overview of the spectrum of security threats and attacks facing computer and network systems. Section 22.1 goes into more detail about the nature of attacks and the types of adversaries that present security threats. In this section, we elaborate on two concepts that are useful in evaluating and clas- sifying threats: attack surfaces and attack trees.

Attack Surfaces

An attack surface consists of the reachable and exploitable vulnerabilities in a sys- tem [MANA11, HOWA03]. Examples of attack surfaces are the following:

■ Open ports on outward facing Web and other servers, and code listening on those ports

■ Services available on the inside of a firewall

■ Code that processes incoming data, email, XML, office documents, and indus- try-specific custom data exchange formats

■ Interfaces, SQL, and Web forms

■ An employee with access to sensitive information vulnerable to a social engineering attack

Attack surfaces can be categorized as follows:

■ Network attack surface: This category refers to vulnerabilities over an enterprise network, wide-area network, or the Internet. Included in this category are net- work protocol vulnerabilities, such as those used for a denial-of-service attack, disruption of communications links, and various forms of intruder attacks.

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■ Software attack surface: This refers to vulnerabilities in application, utility, or operating system code. A particular focus in this category is Web server software.

■ Human attack surface: This category refers to vulnerabilities created by personnel or outsiders, such as social engineering, human error, and trusted insiders.

An attack surface analysis is a useful technique for assessing the scale and severity of threats to a system. A systematic analysis of points of vulnerability makes developers and security analysts aware of where security mechanisms are required. Once an attack surface is defined, designers may be able to find ways to make the surface smaller, thus making the task of the adversary more difficult. The attack surface also provides guidance on setting priorities for testing, strengthening security measures, and modifying the service or application.

As illustrated in Figure 1.3, the use of layering, or defense in depth, and attack surface reduction complement each other in mitigating security risk.

Attack Trees

An attack tree is a branching, hierarchical data structure that represents a set of poten- tial techniques for exploiting security vulnerabilities [MAUW05, MOOR01, SCHN99]. The security incident that is the goal of the attack is represented as the root node of the tree, and the ways that an attacker could reach that goal are iteratively and incre- mentally represented as branches and subnodes of the tree. Each subnode defines a subgoal, and each subgoal may have its own set of further subgoals, and so on. The final nodes on the paths outward from the root, that is, the leaf nodes, represent differ- ent ways to initiate an attack. Each node other than a leaf is either an AND-node or an OR-node. To achieve the goal represented by an AND-node, the subgoals represented by all of that node’s subnodes must be achieved; and for an OR-node, at least one of the subgoals must be achieved. Branches can be labeled with values representing dif- ficulty, cost, or other attack attributes, so that alternative attacks can be compared.

Figure 1.3 Defense in Depth and Attack Surface

Attack surface

Medium security risk

High security risk

Low security riskD

ee p

L ay

er in

g

Sh al

lo w

Small Large

Medium security risk

1.7 / ATTACK SURFACES AND ATTACK TREES 39

The motivation for the use of attack trees is to effectively exploit the infor- mation available on attack patterns. Organizations such as CERT publish security advisories that have enabled the development of a body of knowledge about both general attack strategies and specific attack patterns. Security analysts can use the attack tree to document security attacks in a structured form that reveals key vul- nerabilities. The attack tree can guide both the design of systems and applications, and the choice and strength of countermeasures.

Figure 1.4, based on a figure in [DIMI07], is an example of an attack tree analysis for an Internet banking authentication application. The root of the tree is the objective of the attacker, which is to compromise a user’s account. The shaded boxes on the tree are the leaf nodes, which represent events that comprise the attacks. Note that in this tree, all the nodes other than leaf nodes are OR-nodes. The analysis to generate this tree considered the three components involved in authentication:

Figure 1.4 An Attack Tree for Internet Banking Authentication

Bank account compromise

User credential compromise

User credential guessing

UT/U1a User surveillance

UT/U1b Theft of token and handwritten notes

Malicious software installation Vulnerability exploit

UT/U2a Hidden code

UT/U2b Worms

UT/U3a Smartcard analyzers

UT/U2c Emails with malicious code

UT/U3b Smartcard reader manipulator

UT/U3c Brute force attacks with PIN calculators

CC2 Sniffing

UT/U4a Social engineering

IBS3 Web site manipulation

UT/U4b Web page obfuscation

CC1 Pharming

Redirection of communication toward fraudulent site

CC3 Active man-in-the middle attacks

IBS1 Brute force attacks

User communication with attacker

Injection of commands

Use of known authenticated session by attacker

Normal user authentication with specified session ID

CC4 Pre-defined session IDs (session hijacking)

IBS2 Security policy violation

40 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS

■ User terminal and user (UT/U): These attacks target the user equipment, including the tokens that may be involved, such as smartcards or other pass- word generators, as well as the actions of the user.

■ Communications channel (CC): This type of attack focuses on communica- tion links.

■ Internet banking server (IBS): These types of attacks are offline attacks against the servers that host the Internet banking application.

Five overall attack strategies can be identified, each of which exploits one or more of the three components. The five strategies are as follows:

■ User credential compromise: This strategy can be used against many ele- ments of the attack surface. There are procedural attacks, such as monitoring a user’s action to observe a PIN or other credential, or theft of the user’s token or handwritten notes. An adversary may also compromise token information using a variety of token attack tools, such as hacking the smart- card or using a brute force approach to guess the PIN. Another possible strategy is to embed malicious software to compromise the user’s login and password. An adversary may also attempt to obtain credential information via the communication channel (sniffing). Finally, an adversary may use various means to engage in communication with the target user, as shown in Figure 1.4.

■ Injection of commands: In this type of attack, the attacker is able to intercept communication between the UT and the IBS. Various schemes can be used to be able to impersonate the valid user and so gain access to the banking system.

■ User credential guessing: It is reported in [HILT06] that brute force attacks against some banking authentication schemes are feasible by sending ran- dom usernames and passwords. The attack mechanism is based on distributed zombie personal computers, hosting automated programs for username- or password-based calculation.

■ Security policy violation: For example, violating the bank’s security policy in combination with weak access control and logging mechanisms, an em- ployee may cause an internal security incident and expose a customer’s account.

■ Use of known authenticated session: This type of attack persuades or forces the user to connect to the IBS with a preset session ID. Once the user authen- ticates to the server, the attacker may utilize the known session ID to send packets to the IBS, spoofing the user’s identity.

Figure 1.4 provides a thorough view of the different types of attacks on an Internet banking authentication application. Using this tree as a starting point, secu- rity analysts can assess the risk of each attack and, using the design principles out- lined in the preceding section, design a comprehensive security facility. [DIMO07] provides a good account of the results of this design effort.

1.8 / A MODEL FOR NETWORK SECURITY 41

1.8 A MODEL FOR NETWORK SECURITY

A model for much of what we will be discussing is captured, in very general terms, in Figure 1.5. A message is to be transferred from one party to another across some sort of Internet service. The two parties, who are the principals in this transaction, must cooperate for the exchange to take place. A logical information channel is established by defining a route through the Internet from source to destination and by the coop- erative use of communication protocols (e.g., TCP/IP) by the two principals.

Security aspects come into play when it is necessary or desirable to protect the information transmission from an opponent who may present a threat to confidentiality, authenticity, and so on. All the techniques for providing security have two components:

■ A security-related transformation on the information to be sent. Examples include the encryption of the message, which scrambles the message so that it is unreadable by the opponent, and the addition of a code based on the con- tents of the message, which can be used to verify the identity of the sender.

■ Some secret information shared by the two principals and, it is hoped, unknown to the opponent. An example is an encryption key used in conjunc- tion with the transformation to scramble the message before transmission and unscramble it on reception.6

A trusted third party may be needed to achieve secure transmission. For example, a third party may be responsible for distributing the secret information

6Part Two discusses a form of encryption, known as a symmetric encryption, in which only one of the two principals needs to have the secret information.

Figure 1.5 Model for Network Security

Information channelSecurity-related

transformation

Sender

Secret information

M es

sa ge

M es

sa ge

Se cu

re m

es sa

ge

Se cu

re m

es sa

ge

Recipient

Opponent

Trusted third party (e.g., arbiter, distributer of secret information)

Security-related transformation

Secret information

42 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS

to the two principals while keeping it from any opponent. Or a third party may be needed to arbitrate disputes between the two principals concerning the authenticity of a message transmission.

This general model shows that there are four basic tasks in designing a par- ticular security service:

1. Design an algorithm for performing the security-related transformation. The algorithm should be such that an opponent cannot defeat its purpose.

2. Generate the secret information to be used with the algorithm.

3. Develop methods for the distribution and sharing of the secret information.

4. Specify a protocol to be used by the two principals that makes use of the security algorithm and the secret information to achieve a particular security service.

Parts One through Five of this book concentrate on the types of security mechanisms and services that fit into the model shown in Figure 1.5. However, there are other security-related situations of interest that do not neatly fit this model but are considered in this book. A general model of these other situations is illustrated in Figure 1.6, which reflects a concern for protecting an information system from unwanted access. Most readers are familiar with the concerns caused by the existence of hackers, who attempt to penetrate systems that can be accessed over a network. The hacker can be someone who, with no malign intent, simply gets satisfaction from breaking and entering a computer system. The intruder can be a disgruntled employee who wishes to do damage or a criminal who seeks to exploit computer assets for financial gain (e.g., obtaining credit card numbers or perform- ing illegal money transfers).

Another type of unwanted access is the placement in a computer system of logic that exploits vulnerabilities in the system and that can affect application pro- grams as well as utility programs, such as editors and compilers. Programs can pres- ent two kinds of threats:

■ Information access threats: Intercept or modify data on behalf of users who should not have access to that data.

■ Service threats: Exploit service flaws in computers to inhibit use by legitimate users.

Figure 1.6 Network Access Security Model

Computing resources (processor, memory, I/O)

Data

Processes

Software

Internal security controls

Information system

Gatekeeper function

Opponent —human (e.g., hacker) —software (e.g., virus, worm)

Access channel

1.9 / STANDARDS 43

Viruses and worms are two examples of software attacks. Such attacks can be introduced into a system by means of a disk that contains the unwanted logic con- cealed in otherwise useful software. They can also be inserted into a system across a network; this latter mechanism is of more concern in network security.

The security mechanisms needed to cope with unwanted access fall into two broad categories (see Figure 1.6). The first category might be termed a gatekeeper function. It includes password-based login procedures that are designed to deny access to all but authorized users and screening logic that is designed to detect and reject worms, viruses, and other similar attacks. Once either an unwanted user or unwanted software gains access, the second line of defense consists of a vari- ety of internal controls that monitor activity and analyze stored information in an attempt to detect the presence of unwanted intruders. These issues are explored in Part Six.

1.9 STANDARDS

Many of the security techniques and applications described in this book have been specified as standards. Additionally, standards have been developed to cover man- agement practices and the overall architecture of security mechanisms and services. Throughout this book, we describe the most important standards in use or that are being developed for various aspects of cryptography and network security. Various organizations have been involved in the development or promotion of these stan- dards. The most important (in the current context) of these organizations are as follows:

■ National Institute of Standards and Technology: NIST is a U.S. federal agency that deals with measurement science, standards, and technology related to U.S. government use and to the promotion of U.S. private-sector innovation. Despite its national scope, NIST Federal Information Processing Standards (FIPS) and Special Publications (SP) have a worldwide impact.

■ Internet Society: ISOC is a professional membership society with world- wide organizational and individual membership. It provides leadership in addressing issues that confront the future of the Internet and is the organiza- tion home for the groups responsible for Internet infrastructure standards, including the Internet Engineering Task Force (IETF) and the Internet Architecture Board (IAB). These organizations develop Internet stan- dards and related specifications, all of which are published as Requests for Comments (RFCs).

■ ITU-T: The International Telecommunication Union (ITU) is an interna- tional organization within the United Nations System in which governments and the private sector coordinate global telecom networks and services. The ITU Telecommunication Standardization Sector (ITU-T) is one of the three sectors of the ITU. ITU-T’s mission is the development of technical standards covering all fields of telecommunications. ITU-T standards are referred to as Recommendations.

44 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS

■ ISO: The International Organization for Standardization (ISO)7 is a world- wide federation of national standards bodies from more than 140 countries, one from each country. ISO is a nongovernmental organization that promotes the development of standardization and related activities with a view to fa- cilitating the international exchange of goods and services and to developing cooperation in the spheres of intellectual, scientific, technological, and eco- nomic activity. ISO’s work results in international agreements that are pub- lished as International Standards.

A more detailed discussion of these organizations is contained in Appendix D.

1.10 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS

7ISO is not an acronym (in which case it would be IOS), but it is a word, derived from the Greek, mean- ing equal.

Key Terms

access control active attack authentication authenticity availability data confidentiality data integrity

denial of service encryption integrity intruder masquerade nonrepudiation OSI security architecture

passive attack replay security attacks security mechanisms security services traffic analysis

Review Questions

1.1 What is the OSI security architecture? 1.2 List and briefly define the three key objectives of computer security. 1.3 List and briefly define categories of passive and active security attacks. 1.4 List and briefly define categories of security services. 1.5 List and briefly define categories of security mechanisms. 1.6 List and briefly define the fundamental security design principles. 1.7 Explain the difference between an attack surface and an attack tree.

Problems

1.1 Consider an automated cash deposit machine in which users provide a card or an ac- count number to deposit cash. Give examples of confidentiality, integrity, and avail- ability requirements associated with the system, and, in each case, indicate the degree of importance of the requirement.

1.2 Repeat Problem 1.1 for a payment gateway system where a user pays for an item using their account via the payment gateway.

1.10 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 45

1.3 Consider a financial report publishing system used to produce reports for various organizations. a. Give an example of a type of publication in which confidentiality of the stored

data is the most important requirement. b. Give an example of a type of publication in which data integrity is the most im-

portant requirement. c. Give an example in which system availability is the most important requirement.

1.4 For each of the following assets, assign a low, moderate, or high impact level for the loss of confidentiality, availability, and integrity, respectively. Justify your answers. a. A student maintaining a blog to post public information. b. An examination section of a university that is managing sensitive information

about exam papers. c. An information system in a pathological laboratory maintaining the patient’s data. d. A student information system used for maintaining student data in a university

that contains both personal, academic information and routine administrative in- formation (not privacy related). Assess the impact for the two data sets separately and the information system as a whole.

e. A University library contains a library management system which controls the distribution of books amongst the students of various departments. The library management system contains both the student data and the book data. Assess the impact for the two data sets separately and the information system as a whole.

1.5 Draw a matrix similar to Table 1.4 that shows the relationship between security ser- vices and attacks.

1.6 Draw a matrix similar to Table 1.4 that shows the relationship between security mechanisms and attacks.

1.7 Develop an attack tree for gaining access to the contents of a physical safe. 1.8 Consider a company whose operations are housed in two buildings on the same prop-

erty; one building is headquarters, the other building contains network and computer services. The property is physically protected by a fence around the perimeter, and the only entrance to the property is through this fenced perimeter. In addition to the perimeter fence, physical security consists of a guarded front gate. The local net- works are split between the Headquarters’ LAN and the Network Services’ LAN. Internet users connect to the Web server through a firewall. Dial-up users get access to a particular server on the Network Services’ LAN. Develop an attack tree in which the root node represents disclosure of proprietary secrets. Include physical, social engineering, and technical attacks. The tree may contain both AND and OR nodes. Develop a tree that has at least 15 leaf nodes.

1.9 Read all of the classic papers cited in the Recommended Reading section for this chapter, available at the Author Web site at WilliamStallings.com/Cryptography. The papers are available at box.com/Crypto7e. Compose a 500–1000 word paper (or 8–12 slide PowerPoint presentation) that summarizes the key concepts that emerge from these papers, emphasizing concepts that are common to most or all of the papers.

4646

2.1 Divisibility and The Division Algorithm Divisibility The Division Algorithm

2.2 The Euclidean Algorithm Greatest Common Divisor Finding the Greatest Common Divisor

2.3 Modular Arithmetic The Modulus Properties of Congruences Modular Arithmetic Operations Properties of Modular Arithmetic Euclidean Algorithm Revisited The Extended Euclidean Algorithm

2.4 Prime Numbers

2.5 Fermat’s and Euler’s Theorems

Fermat’s Theorem Euler’s Totient Function Euler’s Theorem

2.6 Testing for Primality

Miller–Rabin Algorithm A Deterministic Primality Algorithm Distribution of Primes

2.7 The Chinese Remainder Theorem

2.8 Discrete Logarithms

The Powers of an Integer, Modulo n Logarithms for Modular Arithmetic Calculation of Discrete Logarithms

2.9 Key Terms, Review Questions, and Problems

Appendix 2A The Meaning of Mod

CHAPTER

Introduction to Number Theory

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2.1 / DIVISIBILITY AND THE DIVISION ALGORITHM 47

Number theory is pervasive in cryptographic algorithms. This chapter provides sufficient breadth and depth of coverage of relevant number theory topics for under- standing the wide range of applications in cryptography. The reader familiar with these topics can safely skip this chapter.

The first three sections introduce basic concepts from number theory that are needed for understanding finite fields; these include divisibility, the Euclidian algo- rithm, and modular arithmetic. The reader may study these sections now or wait until ready to tackle Chapter 5 on finite fields.

Sections 2.4 through 2.8 discuss aspects of number theory related to prime num- bers and discrete logarithms. These topics are fundamental to the design of asymmetric (public-key) cryptographic algorithms. The reader may study these sections now or wait until ready to read Part Three.

The concepts and techniques of number theory are quite abstract, and it is often difficult to grasp them intuitively without examples. Accordingly, this chapter includes a number of examples, each of which is highlighted in a shaded box.

2.1 DIVISIBILITY AND THE DIVISION ALGORITHM

Divisibility

We say that a nonzero b divides a if a = mb for some m, where a, b, and m are integers. That is, b divides a if there is no remainder on division. The notation b � a is commonly used to mean b divides a. Also, if b � a, we say that b is a divisor of a.

LEARNING OBJECTIVES

After studying this chapter, you should be able to:

◆ Understand the concept of divisibility and the division algorithm.

◆ Understand how to use the Euclidean algorithm to find the greatest com- mon divisor.

◆ Present an overview of the concepts of modular arithmetic.

◆ Explain the operation of the extended Euclidean algorithm.

◆ Discuss key concepts relating to prime numbers.

◆ Understand Fermat’s theorem.

◆ Understand Euler’s theorem.

◆ Define Euler’s totient function.

◆ Make a presentation on the topic of testing for primality.

◆ Explain the Chinese remainder theorem.

◆ Define discrete logarithms.

48 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY

Subsequently, we will need some simple properties of divisibility for integers, which are as follows:

■ If a � 1, then a = {1. ■ If a �b and b � a, then a = {b. ■ Any b ≠ 0 divides 0. ■ If a �b and b � c, then a � c:

The positive divisors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.

13 � 182; -5 � 30; 17 � 289; -3 � 33; 17 � 0

11 � 66 and 66 � 198 1 11 � 198

b = 7; g = 14; h = 63; m = 3; n = 2 7 � 14 and 7 � 63. To show 7 � (3 * 14 + 2 * 63), we have (3 * 14 + 2 * 63) = 7(3 * 2 + 2 * 9), and it is obvious that 7 � (7(3 * 2 + 2 * 9)).

■ If b � g and b �h, then b � (mg + nh) for arbitrary integers m and n.

To see this last point, note that

■ If b � g, then g is of the form g = b * g1 for some integer g1. ■ If b �h, then h is of the form h = b * h1 for some integer h1.

So

mg + nh = mbg1 + nbh1 = b * (mg1 + nh1)

and therefore b divides mg + nh.

The Division Algorithm

Given any positive integer n and any nonnegative integer a, if we divide a by n, we get an integer quotient q and an integer remainder r that obey the following relationship:

a = qn + r 0 … r 6 n; q = :a/n; (2.1) where :x; is the largest integer less than or equal to x. Equation (2.1) is referred to as the division algorithm.1

1Equation (2.1) expresses a theorem rather than an algorithm, but by tradition, this is referred to as the division algorithm.

2.2 / THE EUCLIDEAN ALGORITHM 49

Figure 2.1a demonstrates that, given a and positive n, it is always possible to find q and r that satisfy the preceding relationship. Represent the integers on the number line; a will fall somewhere on that line (positive a is shown, a similar dem- onstration can be made for negative a). Starting at 0, proceed to n, 2n, up to qn, such that qn … a and (q + 1)n 7 a. The distance from qn to a is r, and we have found the unique values of q and r. The remainder r is often referred to as a residue.

a = 11; n = 7; 11 = 1 * 7 + 4; r = 4 q = 1 a = -11; n = 7; -11 = (-2) * 7 + 3; r = 3 q = -2

Figure 2.1b provides another example.

Figure 2.1 The Relationship a = qn + r; 0 … r 6 n

0

n 2n 3n qn (q + 1)na

n

r(a) General relationship

0 15

15

10

30 = 2 × 15

70

(b) Example: 70 = (4 × 15) + 10

45 = 3 × 15

60 = 4 × 15

75 = 5 × 15

2.2 THE EUCLIDEAN ALGORITHM

One of the basic techniques of number theory is the Euclidean algorithm, which is a simple procedure for determining the greatest common divisor of two positive integers. First, we need a simple definition: Two integers are relatively prime if and only if their only common positive integer factor is 1.

Greatest Common Divisor

Recall that nonzero b is defined to be a divisor of a if a = mb for some m, where a, b, and m are integers. We will use the notation gcd(a, b) to mean the greatest common divisor of a and b. The greatest common divisor of a and b is the largest integer that divides both a and b. We also define gcd(0, 0) = 0.

50 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY

More formally, the positive integer c is said to be the greatest common divisor of a and b if

1. c is a divisor of a and of b.

2. any divisor of a and b is a divisor of c.

An equivalent definition is the following:

gcd(a, b) = max[k, such that k � a and k �b]

Because we require that the greatest common divisor be positive, gcd(a, b) = gcd(a, -b) = gcd(-a, b) = gcd(-a, -b). In general, gcd(a, b) = gcd( � a � , �b � ).

gcd(60, 24) = gcd(60, -24) = 12

8 and 15 are relatively prime because the positive divisors of 8 are 1, 2, 4, and 8, and the positive divisors of 15 are 1, 3, 5, and 15. So 1 is the only integer on both lists.

Also, because all nonzero integers divide 0, we have gcd(a, 0) = � a � . We stated that two integers a and b are relatively prime if and only if their

only common positive integer factor is 1. This is equivalent to saying that a and b are relatively prime if gcd(a, b) = 1.

Finding the Greatest Common Divisor

We now describe an algorithm credited to Euclid for easily finding the greatest common divisor of two integers (Figure 2.2). This algorithm has broad significance in cryptography. The explanation of the algorithm can be broken down into the fol- lowing points:

1. Suppose we wish to determine the greatest common divisor d of the integers a and b; that is determine d = gcd(a, b). Because gcd( � a � , �b � ) = gcd(a, b), there is no harm in assuming a Ú b 7 0.

2. Dividing a by b and applying the division algorithm, we can state:

a = q1b + r1 0 … r1 6 b (2.2)

3. First consider the case in which r1 = 0. Therefore b divides a and clearly no larger number divides both b and a, because that number would be larger than b. So we have d = gcd(a, b) = b.

4. The other possibility from Equation (2.2) is r1 ≠ 0. For this case, we can state that d � r1. This is due to the basic properties of divisibility: the relations d � a and d �b together imply that d � (a – q1b), which is the same as d � r1.

5. Before proceeding with the Euclidian algorithm, we need to answer the ques- tion: What is the gcd(b, r1)? We know that d �b and d � r1. Now take any arbi- trary integer c that divides both b and r1. Therefore, c � (q1b + r1) = a. Because c divides both a and b, we must have c … d, which is the greatest common divisor of a and b. Therefore d = gcd(b, r1).

2.2 / THE EUCLIDEAN ALGORITHM 51

Let us now return to Equation (2.2) and assume that r1 ≠ 0. Because b 7 r1, we can divide b by r1 and apply the division algorithm to obtain:

b = q2r1 + r2 0 … r2 6 r1

As before, if r2 = 0, then d = r1 and if r2 ≠ 0, then d = gcd(r1, r2). Note that the remainders form a descending series of nonnegative values and so must terminate when the remainder is zero. This happens, say, at the (n + 1)th stage where rn – 1 is divided by rn. The result is the following system of equations:

a = q1b + r1 0 6 r1 6 b b = q2r1 + r2 0 6 r2 6 r1 r1 = q3r2 + r3 0 6 r3 6 r2

~ ~

~ ~ ~ ~

rn – 2 = qnrn – 1 + rn 0 6 rn 6 rn – 1 rn – 1 = qn + 1rn + 0 d = gcd(a, b) = rn

w (2.3) At each iteration, we have d = gcd(ri, ri+ 1) until finally d = gcd(rn, 0) = rn.

Thus, we can find the greatest common divisor of two integers by repetitive appli- cation of the division algorithm. This scheme is known as the Euclidean algorithm. Figure 2.3 illustrates a simple example.

We have essentially argued from the top down that the final result is the gcd(a, b). We can also argue from the bottom up. The first step is to show that rn divides a and b. It follows from the last division in Equation (2.3) that rn divides rn – 1. The next to last division shows that rn divides rn – 2 because it divides both

Figure 2.2 Euclidean Algorithm

No

No Yes a > b?

r > 0? Swap

a and b

Replace b with r

Replace a with b

Divide a by b, calling the

remainder r

GCD is the final

value of b

START

END Figure 2.3 Euclidean Algorithm Example: gcd(710, 310)

710 = 2 × 310 + 90

310 = 3 × 90 + 40

90 = 2 × 40 + 10

40 = 4 × 10

GCDGCD

Same GCD

52 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY

terms on the right. Successively, one sees that rn divides all ri>s and finally a and b. It remains to show that rn is the largest divisor that divides a and b. If we take any arbitrary integer that divides a and b, it must also divide r1, as explained previously. We can follow the sequence of equations in Equation (2.3) down and show that c must divide all ri>s. Therefore c must divide rn, so that rn = gcd(a, b).

Let us now look at an example with relatively large numbers to see the power of this algorithm:

To find d = gcd(a, b) = gcd(1160718174, 316258250)

a = q1b + r1 1160718174 = 3 * 316258250 + 211943424 d = gcd(316258250, 211943424) b = q2r1 + r2 316258250 = 1 * 211943424 + 104314826 d = gcd(211943424, 104314826) r1 = q3r2 + r3 211943424 = 2 * 104314826 + 3313772 d = gcd(104314826, 3313772) r2 = q4r3 + r4 104314826 = 31 * 3313772 + 1587894 d = gcd(3313772, 1587894) r3 = q5r4 + r5 3313772 = 2 * 1587894 + 137984 d = gcd(1587894, 137984) r4 = q6r5 + r6 1587894 = 11 * 137984 + 70070 d = gcd(137984, 70070) r5 = q7r6 + r7 137984 = 1 * 70070 + 67914 d = gcd(70070, 67914) r6 = q8r7 + r8 70070 = 1 * 67914 + 2156 d = gcd(67914, 2156) r7 = q9r8 + r9 67914 = 31 * 2156 + 1078 d = gcd(2156, 1078) r8 = q10r9 + r10 2156 = 2 * 1078 + 0 d = gcd(1078, 0) = 1078 Therefore, d = gcd(1160718174, 316258250) = 1078

In this example, we begin by dividing 1160718174 by 316258250, which gives 3 with a remainder of 211943424. Next we take 316258250 and divide it by 211943424. The process continues until we get a remainder of 0, yielding a result of 1078.

It will be helpful in what follows to recast the above computation in tabular form. For every step of the iteration, we have ri- 2 = qiri- 1 + ri, where ri- 2 is the dividend, ri- 1 is the divisor, qi is the quotient, and ri is the remainder. Table 2.1 sum- marizes the results.

Dividend Divisor Quotient Remainder

a = 1160718174 b = 316258250 q1 = 3 r1 = 211943424

b = 316258250 r1 = 211943434 q2 = 1 r2 = 104314826

r1 = 211943424 r2 = 104314826 q3 = 2 r3 = 3313772

r2 = 104314826 r3 = 3313772 q4 = 31 r4 = 1587894

r3 = 3313772 r4 = 1587894 q5 = 2 r5 = 137984

r4 = 1587894 r5 = 137984 q6 = 11 r6 = 70070

r5 = 137984 r6 = 70070 q7 = 1 r7 = 67914

r6 = 70070 r7 = 67914 q8 = 1 r8 = 2156

r7 = 67914 r8 = 2156 q9 = 31 r9 = 1078

r8 = 2156 r9 = 1078 q10 = 2 r10 = 0

Table 2.1 Euclidean Algorithm Example

2.3 / MODULAR ARITHMETIC 53

2.3 MODULAR ARITHMETIC

The Modulus

If a is an integer and n is a positive integer, we define a mod n to be the remainder when a is divided by n. The integer n is called the modulus. Thus, for any integer a, we can rewrite Equation (2.1) as follows:

a = qn + r 0 … r 6 n; q = :a/n; a = :a/n; * n + (a mod n)

11 mod 7 = 4; -11 mod 7 = 3

73 K 4 (mod 23); 21 K -9 (mod 10)

Two integers a and b are said to be congruent modulo n, if (a mod n) = (b mod n). This is written as a K b (mod n).2

2We have just used the operator mod in two different ways: first as a binary operator that produces a re- mainder, as in the expression a mod b; second as a congruence relation that shows the equivalence of two integers, as in the expression a K b (mod n). See Appendix 2A for a discussion.

Note that if a K 0 (mod n), then n � a.

Properties of Congruences

Congruences have the following properties:

1. a K b (mod n) if n � (a – b). 2. a K b (mod n) implies b K a (mod n). 3. a K b (mod n) and b K c (mod n) imply a K c (mod n).

To demonstrate the first point, if n � (a – b), then (a – b) = kn for some k. So we can write a = b + kn. Therefore, (a mod n) = (remainder when b + kn is divided by n) = (remainder when b is divided by n) = (b mod n).

23 K 8 (mod 5) because 23 – 8 = 15 = 5 * 3 -11 K 5 (mod 8) because -11 – 5 = -16 = 8 * (-2) 81 K 0 (mod 27) because 81 – 0 = 81 = 27 * 3

The remaining points are as easily proved.

54 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY

Modular Arithmetic Operations

Note that, by definition (Figure 2.1), the (mod n) operator maps all integers into the set of integers {0, 1, c , (n – 1)}. This suggests the question: Can we perform arithmetic operations within the confines of this set? It turns out that we can; this technique is known as modular arithmetic.

Modular arithmetic exhibits the following properties:

1. [(a mod n) + (b mod n)] mod n = (a + b) mod n 2. [(a mod n) – (b mod n)] mod n = (a – b) mod n 3. [(a mod n) * (b mod n)] mod n = (a * b) mod n

We demonstrate the first property. Define (a mod n) = ra and (b mod n) = rb. Then we can write a = ra + jn for some integer j and b = rb + kn for some integer k. Then

(a + b) mod n = (ra + jn + rb + kn) mod n = (ra + rb + (k + j)n) mod n = (ra + rb) mod n = [(a mod n) + (b mod n)] mod n

The remaining properties are proven as easily. Here are examples of the three properties:

11 mod 8 = 3; 15 mod 8 = 7 [(11 mod 8) + (15 mod 8)] mod 8 = 10 mod 8 = 2 (11 + 15) mod 8 = 26 mod 8 = 2 [(11 mod 8) – (15 mod 8)] mod 8 = -4 mod 8 = 4 (11 – 15) mod 8 = -4 mod 8 = 4 [(11 mod 8) * (15 mod 8)] mod 8 = 21 mod 8 = 5 (11 * 15) mod 8 = 165 mod 8 = 5

To find 117 mod 13, we can proceed as follows:

112 = 121 K 4 (mod 13) 114 = (112)2 K 42 K 3 (mod 13) 117 = 11 * 112 * 114

117 K 11 * 4 * 3 K 132 K 2 (mod 13)

Exponentiation is performed by repeated multiplication, as in ordinary arithmetic.

Thus, the rules for ordinary arithmetic involving addition, subtraction, and multiplication carry over into modular arithmetic.

2.3 / MODULAR ARITHMETIC 55

Table 2.2 provides an illustration of modular addition and multiplication modulo 8. Looking at addition, the results are straightforward, and there is a reg- ular pattern to the matrix. Both matrices are symmetric about the main diagonal in conformance to the commutative property of addition and multiplication. As in ordinary addition, there is an additive inverse, or negative, to each integer in modu- lar arithmetic. In this case, the negative of an integer x is the integer y such that (x + y) mod 8 = 0. To find the additive inverse of an integer in the left-hand col- umn, scan across the corresponding row of the matrix to find the value 0; the integer at the top of that column is the additive inverse; thus, (2 + 6) mod 8 = 0. Similarly, the entries in the multiplication table are straightforward. In modular arithmetic mod 8, the multiplicative inverse of x is the integer y such that (x * y) mod 8 = 1 mod 8. Now, to find the multiplicative inverse of an integer from the multiplication table, scan across the matrix in the row for that integer to find the value 1; the integer at the top of that column is the multiplicative inverse; thus, (3 * 3) mod 8 = 1. Note that not all integers mod 8 have a multiplicative inverse; more about that later.

Properties of Modular Arithmetic

Define the set Zn as the set of nonnegative integers less than n:

Zn = {0, 1, c , (n – 1)}

Table 2.2 Arithmetic Modulo 8 + 0 1 2 3 4 5 6 7

0 0 1 2 3 4 5 6 7

1 1 2 3 4 5 6 7 0

2 2 3 4 5 6 7 0 1

3 3 4 5 6 7 0 1 2

4 4 5 6 7 0 1 2 3

5 5 6 7 0 1 2 3 4

6 6 7 0 1 2 3 4 5

7 7 0 1 2 3 4 5 6

(a) Addition modulo 8

* 0 1 2 3 4 5 6 7

0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7

2 0 2 4 6 0 2 4 6

3 0 3 6 1 4 7 2 5

4 0 4 0 4 0 4 0 4

5 0 5 2 7 4 1 6 3

6 0 6 4 2 0 6 4 2

7 0 7 6 5 4 3 2 1

(b) Multiplication modulo 8

w -w w-1

0 0 —

1 7 1

2 6 —

3 5 3

4 4 —

5 3 5

6 2 —

7 1 7

(c) Additive and multiplicative inverse modulo 8

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http://www.hiva-network.com/

56 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY

This is referred to as the set of residues, or residue classes (mod n). To be more pre- cise, each integer in Zn represents a residue class. We can label the residue classes (mod n) as [0], [1], [2], c , [n – 1], where

[r] = {a: a is an integer, a K r (mod n)}

The residue classes (mod 4) are

[0] = {c , -16, -12, -8, -4, 0, 4, 8, 12, 16, c } [1] = {c , -15, -11, -7, -3, 1, 5, 9, 13, 17, c } [2] = {c , -14, -10, -6, -2, 2, 6, 10, 14, 18, c } [3] = {c , -13, -9, -5, -1, 3, 7, 11, 15, 19, c }

Property Expression

Commutative Laws (w + x) mod n = (x + w) mod n (w * x) mod n = (x * w) mod n

Associative Laws [(w + x) + y] mod n = [w + (x + y)] mod n [(w * x) * y] mod n = [w * (x * y)] mod n

Distributive Law [w * (x + y)] mod n = [(w * x) + (w * y)] mod n

Identities (0 + w) mod n = w mod n (1 * w) mod n = w mod n

Additive Inverse (-w) For each w∈ Zn, there exists a z such that w + z K 0 mod n

Table 2.3 Properties of Modular Arithmetic for Integers in Zn

Of all the integers in a residue class, the smallest nonnegative integer is the one used to represent the residue class. Finding the smallest nonnegative integer to which k is congruent modulo n is called reducing k modulo n.

If we perform modular arithmetic within Zn, the properties shown in Table 2.3 hold for integers in Zn. We show in the next section that this implies that Zn is a commutative ring with a multiplicative identity element.

There is one peculiarity of modular arithmetic that sets it apart from ordinary arithmetic. First, observe that (as in ordinary arithmetic) we can write the following:

if (a + b) K (a + c) (mod n) then b K c (mod n) (2.4)

(5 + 23) K (5 + 7)(mod 8); 23 K 7(mod 8)

Equation (2.4) is consistent with the existence of an additive inverse. Adding the additive inverse of a to both sides of Equation (2.4), we have

((-a) + a + b) K ((-a) + a + c)(mod n) b K c (mod n)

2.3 / MODULAR ARITHMETIC 57

However, the following statement is true only with the attached condition:

if (a * b) K (a * c)(mod n) then b K c(mod n) if a is relatively prime to n (2.5)

Recall that two integers are relatively prime if their only common positive integer factor is 1. Similar to the case of Equation (2.4), we can say that Equation (2.5) is consistent with the existence of a multiplicative inverse. Applying the multiplicative inverse of a to both sides of Equation (2.5), we have

((a-1)ab) K ((a-1)ac)(mod n) b K c(mod n)

To see this, consider an example in which the condition of Equation (2.5) does not hold. The integers 6 and 8 are not relatively prime, since they have the common factor 2. We have the following:

6 * 3 = 18 K 2(mod 8) 6 * 7 = 42 K 2(mod 8)

Yet 3 [ 7 (mod 8).

The reason for this strange result is that for any general modulus n, a multi- plier a that is applied in turn to the integers 0 through (n – 1) will fail to produce a complete set of residues if a and n have any factors in common.

With a = 6 and n = 8,

Z8 0 1 2 3 4 5 6 7 Multiply by 6 0 6 12 18 24 30 36 42 Residues 0 6 4 2 0 6 4 2

Because we do not have a complete set of residues when multiplying by 6, more than one integer in Z8 maps into the same residue. Specifically, 6 * 0 mod 8 = 6 * 4 mod 8; 6 * 1 mod 8 = 6 * 5 mod 8; and so on. Because this is a many-to-one mapping, there is not a unique inverse to the multiply operation.

However, if we take a = 5 and n = 8, whose only common factor is 1,

Z8 0 1 2 3 4 5 6 7 Multiply by 5 0 5 10 15 20 25 30 35 Residues 0 5 2 7 4 1 6 3

The line of residues contains all the integers in Z8, in a different order.

58 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY

In general, an integer has a multiplicative inverse in Zn if and only if that inte- ger is relatively prime to n. Table 2.2c shows that the integers 1, 3, 5, and 7 have a multiplicative inverse in Z8; but 2, 4, and 6 do not.

Euclidean Algorithm Revisited

The Euclidean algorithm can be based on the following theorem: For any integers a, b, with a Ú b Ú 0,

gcd(a, b) = gcd(b, a mod b) (2.6)

gcd(55, 22) = gcd(22, 55 mod 22) = gcd(22, 11) = 11

gcd(18, 12) = gcd(12, 6) = gcd(6, 0) = 6 gcd(11, 10) = gcd(10, 1) = gcd(1, 0) = 1

To see that Equation (2.6) works, let d = gcd(a, b). Then, by the definition of gcd, d � a and d �b. For any positive integer b, we can express a as

a = kb + r K r (mod b) a mod b = r

with k, r integers. Therefore, (a mod b) = a – kb for some integer k. But because d �b, it also divides kb. We also have d � a. Therefore, d � (a mod b). This shows that d is a common divisor of b and (a mod b). Conversely, if d is a common divisor of b and (a mod b), then d �kb and thus d � [kb + (a mod b)], which is equivalent to d � a. Thus, the set of common divisors of a and b is equal to the set of common divisors of b and (a mod b). Therefore, the gcd of one pair is the same as the gcd of the other pair, proving the theorem.

Equation (2.6) can be used repetitively to determine the greatest common divisor.

This is the same scheme shown in Equation (2.3), which can be rewritten in the following way.

Euclidean Algorithm

Calculate Which satisfies

r1 = a mod b a = q1b + r1 r2 = b mod r1 b = q2r1 + r2 r3 = r1 mod r2 r1 = q3r2 + r3

~

~

~

~

~

~

rn = rn – 2 mod rn – 1 rn – 2 = qnrn – 1 + rn rn + 1 = rn – 1 mod rn = 0 rn – 1 = qn + 1rn + 0

d = gcd(a, b) = rn

We can define the Euclidean algorithm concisely as the following recursive function.

2.3 / MODULAR ARITHMETIC 59

Euclid(a,b) if (b=0) then return a; else return Euclid(b, a mod b);

The Extended Euclidean Algorithm

We now proceed to look at an extension to the Euclidean algorithm that will be important for later computations in the area of finite fields and in encryption algo- rithms, such as RSA. For given integers a and b, the extended Euclidean algorithm not only calculates the greatest common divisor d but also two additional integers x and y that satisfy the following equation.

ax + by = d = gcd(a, b) (2.7)

It should be clear that x and y will have opposite signs. Before examining the algorithm, let us look at some of the values of x and y when a = 42 and b = 30. Note that gcd(42, 30) = 6. Here is a partial table of values3 for 42x + 30y.

x − 3 − 2 − 1 0 1 2 3

y

-3 -216 -174 -132 -90 -48 -6 36 -2 -186 -144 -102 -60 -18 24 66 -1 -156 -114 -72 -30 12 54 96

0 -126 -84 -42 0 42 84 126 1 -96 -54 -12 30 72 114 156 2 -66 -24 18 60 102 144 186 3 -36 6 48 90 132 174 216

Observe that all of the entries are divisible by 6. This is not surpris- ing, because both 42 and 30 are divisible by 6, so every number of the form 42x + 30y = 6(7x + 5y) is a multiple of 6. Note also that gcd(42, 30) = 6 appears in the table. In general, it can be shown that for given integers a and b, the smallest positive value of ax + by is equal to gcd(a, b).

Now let us show how to extend the Euclidean algorithm to determine (x, y, d) given a and b. We again go through the sequence of divisions indicated in Equation (2.3), and we assume that at each step i we can find integers xi and yi that satisfy ri = axi + byi. We end up with the following sequence.

a = q1b + r1 r1 = ax1 + by1 b = q2r1 + r2 r2 = ax2 + by2 r1 = q3r2 + r3 r3 = ax3 + by3

f f rn – 2 = qnrn – 1 + rn rn = axn + byn rn – 1 = qn + 1rn + 0

3This example is taken from [SILV06].

60 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY

Now, observe that we can rearrange terms to write

ri = ri- 2 – ri- 1qi (2.8)

Also, in rows i – 1 and i – 2, we find the values

ri- 2 = axi- 2 + byi- 2 and ri- 1 = axi- 1 + byi- 1

Substituting into Equation (2.8), we have

ri = (axi- 2 + byi- 2) – (axi- 1 + byi- 1)qi = a(xi- 2 – qixi- 1) + b(yi- 2 – qiyi- 1)

But we have already assumed that ri = axi + byi. Therefore,

xi = xi- 2 – qixi- 1 and yi = yi- 2 – qiyi- 1

We now summarize the calculations:

Extended Euclidean Algorithm

Calculate Which satisfies Calculate Which satisfies

r-1 = a x-1 = 1; y-1 = 0 a = ax-1 + by-1 r0 = b x0 = 0; y0 = 1 b = ax0 + by0 r1 = a mod b q1 = :a/b;

a = q1b + r1 x1 = x-1 – q1x0 = 1 y1 = y-1 – q1y0 = -q1

r1 = ax1 + by1

r2 = b mod r1 q2 = :b/r1;

b = q2r1 + r2 x2 = x0 – q2x1 y2 = y0 – q2y1

r2 = ax2 + by2

r3 = r1 mod r2 q3 = :r1/r2;

r1 = q3r2 + r3 x3 = x1 – q3x2 y3 = y1 – q3y2

r3 = ax3 + by3

~

~

~

~

~

~

~

~

~

~

~

~

rn = rn – 2 mod rn – 1 qn = :rn – 2/rn – 1;

rn – 2 = qnrn – 1 + rn xn = xn – 2 – qnxn – 1 yn = yn – 2 – qnyn – 1

rn = axn + byn

rn + 1 = rn – 1 mod rn = 0 qn + 1 = :rn – 1/rn;

rn – 1 = qn + 1rn + 0 d = gcd(a, b) = rn x = xn; y = yn

We need to make several additional comments here. In each row, we calculate a new remainder ri based on the remainders of the previous two rows, namely ri- 1 and ri- 2. To start the algorithm, we need values for r0 and r-1, which are just a and b. It is then straightforward to determine the required values for x-1, y-1, x0, and y0.

We know from the original Euclidean algorithm that the process ends with a remainder of zero and that the greatest common divisor of a and b is d = gcd(a, b) = rn. But we also have determined that d = rn = axn + byn. Therefore, in Equation (2.7), x = xn and y = yn.

As an example, let us use a = 1759 and b = 550 and solve for 1759x + 550y = gcd(1759, 550). The results are shown in Table 2.4. Thus, we have 1759 * (-111) + 550 * 355 = -195249 + 195250 = 1.

2.4 / PRIME NUMBERS 61

2.4 PRIME NUMBERS4

A central concern of number theory is the study of prime numbers. Indeed, whole books have been written on the subject (e.g., [CRAN01], [RIBE96]). In this section, we provide an overview relevant to the concerns of this book.

An integer p 7 1 is a prime number if and only if its only divisors5 are {1 and {p. Prime numbers play a critical role in number theory and in the techniques dis- cussed in this chapter. Table 2.5 shows the primes less than 2000. Note the way the primes are distributed. In particular, note the number of primes in each range of 100 numbers.

Any integer a 7 1 can be factored in a unique way as

a = p1a1 * p2a2 * g * ptat (2.9)

where p1 6 p2 6 c 6 pt are prime numbers and where each ai is a positive inte- ger. This is known as the fundamental theorem of arithmetic; a proof can be found in any text on number theory.

4In this section, unless otherwise noted, we deal only with the nonnegative integers. The use of negative integers would introduce no essential differences. 5Recall from Section 2.1 that integer a is said to be a divisor of integer b if there is no remainder on division. Equivalently, we say that a divides b.

i ri qi xi yi

-1 1759 1 0

0 550 0 1

1 109 3 1 -3

2 5 5 -5 16

3 4 21 106 -339

4 1 1 -111 355

5 0 4

Result: d = 1; x = -111; y = 355

Table 2.4 Extended Euclidean Algorithm Example

91 = 7 * 13 3600 = 24 * 32 * 52

11011 = 7 * 112 * 13

It is useful for what follows to express this another way. If P is the set of all prime numbers, then any positive integer a can be written uniquely in the following form:

a = q p∈P

pap where each ap Ú 0

62 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY

2 10

1 21

1 30

7 40

1 50

3 60

1 70

1 80

9 90

7 10

09 11

03 12

01 13

01 14

09 15

11 16

01 17

09 18

01 19

01

3 10

3 22

3 31

1 40

9 50

9 60

7 70

9 81

1 91

1 10

13 11

09 12

13 13

03 14

23 15

23 16

07 17

21 18

11 19

07

5 10

7 22

7 31

3 41

9 52

1 61

3 71

9 82

1 91

9 10

19 11

17 12

17 13

07 14

27 15

31 16

09 17

23 18

23 19

13

7 10

9 22

9 31

7 42

1 52

3 61

7 72

7 82

3 92

9 10

21 11

23 12

23 13

19 14

29 15

43 16

13 17

33 18

31 19

31

11 11

3 23

3 33

1 43

1 54

1 61

9 73

3 82

7 93

7 10

31 11

29 12

29 13

21 14

33 15

49 16

19 17

41 18

47 19

33

13 12

7 23

9 33

7 43

3 54

7 63

1 73

9 82

9 94

1 10

33 11

51 12

31 13

27 14

39 15

53 16

21 17

47 18

61 19

49

17 13

1 24

1 34

7 43

9 55

7 64

1 74

3 83

9 94

7 10

39 11

53 12

37 13

61 14

47 15

59 16

27 17

53 18

67 19

51

19 13

7 25

1 34

9 44

3 56

3 64

3 75

1 85

3 95

3 10

49 11

63 12

49 13

67 14

51 15

67 16

37 17

59 18

71 19

73

23 13

9 25

7 35

3 44

9 56

9 64

7 75

7 85

7 96

7 10

51 11

71 12

59 13

73 14

53 15

71 16

57 17

77 18

73 19

79

29 14

9 26

3 35

9 45

7 57

1 65

3 76

1 85

9 97

1 10

61 11

81 12

77 13

81 14

59 15

79 16

63 17

83 18

77 19

87

31 15

1 26

9 36

7 46

1 57

7 65

9 76

9 86

3 97

7 10

63 11

87 12

79 13

99 14

71 15

83 16

67 17

87 18

79 19

93

37 15

7 27

1 37

3 46

3 58

7 66

1 77

3 87

7 98

3 10

69 11

93 12

83 14

81 15

97 16

69 17

89 18

89 19

97

41 16

3 27

7 37

9 46

7 59

3 67

3 78

7 88

1 99

1 10

87 12

89 14

83 16

93 19

99

43 16

7 28

1 38

3 47

9 59

9 67

7 79

7 88

3 99

7 10

91 12

91 14

87 16

97

47 17

3 28

3 38

9 48

7 68

3 88

7 10

93 12

97 14

89 16

99

53 17

9 29

3 39

7 49

1 69

1 10

97 14

93

59 18

1 49

9 14

99

61 19

1

67 19

3

71 19

7

73 19

9

79 83 89 97

T ab

le 2

.5

P ri

m es

U nd

er 2

00 0

2.4 / PRIME NUMBERS 63

The right-hand side is the product over all possible prime numbers p; for any par- ticular value of a, most of the exponents ap will be 0.

The value of any given positive integer can be specified by simply listing all the nonzero exponents in the foregoing formulation.

The integer 12 is represented by {a2 = 2, a3 = 1}. The integer 18 is represented by {a2 = 1, a3 = 2}. The integer 91 is represented by {a7 = 1, a13 = 1}.

Multiplication of two numbers is equivalent to adding the corresponding

exponents. Given a = q p∈P

pap, b = q p∈P

pbp. Define k = ab. We know that the inte-

ger k can be expressed as the product of powers of primes: k = q p∈P

pkp. It follows that kp = ap + bp for all p ∈ P.

k = 12 * 18 = (22 * 3) * (2 * 32) = 216 k2 = 2 + 1 = 3; k3 = 1 + 2 = 3 216 = 23 * 33 = 8 * 27

a = 12; b = 36; 12 � 36 12 = 22 * 3; 36 = 22 * 32

a2 = 2 = b2 a3 = 1 … 2 = b3 Thus, the inequality ap … bp is satisfied for all prime numbers.

What does it mean, in terms of the prime factors of a and b, to say that a divides b? Any integer of the form pn can be divided only by an integer that is of a lesser or equal power of the same prime number, pj with j … n. Thus, we can say the following.

Given

a = q p∈P

pap, b = q p∈P

pbp

If a �b, then ap … bp for all p.

It is easy to determine the greatest common divisor of two positive integers if we express each integer as the product of primes.

64 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY

The following relationship always holds:

If k = gcd(a, b), then kp = min(ap, bp) for all p.

Determining the prime factors of a large number is no easy task, so the pre- ceding relationship does not directly lead to a practical method of calculating the greatest common divisor.

2.5 FERMAT’S AND EULER’S THEOREMS

Two theorems that play important roles in public-key cryptography are Fermat’s theorem and Euler’s theorem.

Fermat’s Theorem6

Fermat’s theorem states the following: If p is prime and a is a positive integer not divisible by p, then

ap – 1 K 1 (mod p) (2.10)

Proof: Consider the set of positive integers less than p: {1, 2, c , p – 1} and mul- tiply each element by a, modulo p, to get the set X = {a mod p, 2a mod p, c , (p – 1)a mod p}. None of the elements of X is equal to zero because p does not divide a. Furthermore, no two of the integers in X are equal. To see this, assume that ja K ka(mod p)), where 1 … j 6 k … p – 1. Because a is relatively prime7 to p, we can eliminate a from both sides of the equation [see Equation (2.3)] resulting in j K k(mod p). This last equality is impossible, because j and k are both positive inte- gers less than p. Therefore, we know that the (p – 1) elements of X are all positive integers with no two elements equal. We can conclude the X consists of the set of integers {1, 2, c , p – 1} in some order. Multiplying the numbers in both sets (p and X) and taking the result mod p yields

a * 2a * g * (p – 1)a K [(1 * 2 * g * (p – 1)](mod p) ap – 1(p – 1)! K (p – 1)! (mod p)

We can cancel the (p – 1)! term because it is relatively prime to p [see Equation (2.5)]. This yields Equation (2.10), which completes the proof.

6This is sometimes referred to as Fermat’s little theorem. 7Recall from Section 2.2 that two numbers are relatively prime if they have no prime factors in common; that is, their only common divisor is 1. This is equivalent to saying that two numbers are relatively prime if their greatest common divisor is 1.

300 = 22 * 31 * 52

18 = 21 * 32

gcd(18,300) = 21 * 31 * 50 = 6

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2.5 / FERMAT’S AND EULER’S THEOREMS 65

An alternative form of Fermat’s theorem is also useful: If p is prime and a is a positive integer, then

ap K a(mod p) (2.11)

Note that the first form of the theorem [Equation (2.10)] requires that a be rela- tively prime to p, but this form does not.

a = 7, p = 19 72 = 49 K 11 (mod 19) 74 K 121 K 7 (mod 19) 78 K 49 K 11 (mod 19) 716 K 121 K 7 (mod 19) ap – 1 = 718 = 716 * 72 K 7 * 11 K 1 (mod 19)

p = 5, a = 3 ap = 35 = 243 K 3(mod 5) = a(mod p) p = 5, a = 10 ap = 105 = 100000 K 10(mod 5) K 0(mod 5) = a(mod p)

Euler’s Totient Function

Before presenting Euler’s theorem, we need to introduce an important quantity in number theory, referred to as Euler’s totient function. This function, written f(n), is defined as the number of positive integers less than n and relatively prime to n. By convention, f(1) = 1.

Determine f(37) and f(35).

Because 37 is prime, all of the positive integers from 1 through 36 are relatively prime to 37. Thus f(37) = 36. To determine f(35), we list all of the positive integers less than 35 that are relatively prime to it:

1, 2, 3, 4, 6, 8, 9, 11, 12, 13, 16, 17, 18

19, 22, 23, 24, 26, 27, 29, 31, 32, 33, 34

There are 24 numbers on the list, so f(35) = 24.

Table 2.6 lists the first 30 values of f(n). The value f(1) is without meaning but is defined to have the value 1.

It should be clear that, for a prime number p,

f(p) = p – 1

Now suppose that we have two prime numbers p and q with p ≠ q. Then we can show that, for n = pq,

66 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY

f(n) = f(pq) = f(p) * f(q) = (p – 1) * (q – 1)

To see that f(n) = f(p) * f(q), consider that the set of positive integers less than n is the set {1, c , (pq – 1)}. The integers in this set that are not relatively prime to n are the set {p, 2p, c , (q – 1)p} and the set {q, 2q, c , (p – 1)q}. To see this, consider that any integer that divides n must divide either of the prime num- bers p or q. Therefore, any integer that does not contain either p or q as a factor is relatively prime to n. Further note that the two sets just listed are non-overlapping: Because p and q are prime, we can state that none of the integers in the first set can be written as a multiple of q, and none of the integers in the second set can be writ- ten as a multiple of p. Thus the total number of unique integers in the two sets is (q – 1) + (p – 1). Accordingly,

f(n) = (pq – 1) – [(q – 1) + (p – 1)] = pq – (p + q) + 1 = (p – 1) * (q – 1) = f(p) * f(q)

f(21) = f(3) * f(7) = (3 – 1) * (7 – 1) = 2 * 6 = 12 where the 12 integers are {1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20}.

Table 2.6 Some Values of Euler’s Totient Function f(n)

n f(n)

1 1

2 1

3 2

4 2

5 4

6 2

7 6

8 4

9 6

10 4

n f(n)

11 10

12 4

13 12

14 6

15 8

16 8

17 16

18 6

19 18

20 8

n f(n)

21 12

22 10

23 22

24 8

25 20

26 12

27 18

28 12

29 28

30 8

Euler’s Theorem

Euler’s theorem states that for every a and n that are relatively prime:

af(n) K 1(mod n) (2.12)

Proof: Equation (2.12) is true if n is prime, because in that case, f(n) = (n – 1) and Fermat’s theorem holds. However, it also holds for any integer n. Recall that

2.5 / FERMAT’S AND EULER’S THEOREMS 67

f(n) is the number of positive integers less than n that are relatively prime to n. Consider the set of such integers, labeled as

R = {x1, x2, c , xf(n)}

That is, each element xi of R is a unique positive integer less than n with gcd(xi, n) = 1. Now multiply each element by a, modulo n:

S = {(ax1 mod n), (ax2 mod n), c , (axf(n) mod n)}

The set S is a permutation8 of R , by the following line of reasoning:

1. Because a is relatively prime to n and xi is relatively prime to n, axi must also be relatively prime to n. Thus, all the members of S are integers that are less than n and that are relatively prime to n.

2. There are no duplicates in S. Refer to Equation (2.5). If axi mod n= axj mod n, then xi = xj.

Therefore,

q f(n)

i=1 (axi mod n) = q

f(n)

i=1 xi

q f(n)

i=1 axi K q

f(n)

i=1 xi (mod n)

af(n) * Jqf(n) i=1

xiR K qf(n) i=1

xi (mod n)

af(n) K 1 (mod n)

which completes the proof. This is the same line of reasoning applied to the proof of Fermat’s theorem.

8A permutation of a finite set of elements S is an ordered sequence of all the elements of S, with each element appearing exactly once.

a = 3; n = 10; f(10) = 4; af(n) = 34 = 81 = 1(mod 10) = 1(mod n) a = 2; n = 11; f(11) = 10; af(n) = 210 = 1024 = 1(mod 11) = 1(mod n)

As is the case for Fermat’s theorem, an alternative form of the theorem is also useful:

af(n) + 1 K a(mod n) (2.13)

Again, similar to the case with Fermat’s theorem, the first form of Euler’s theorem [Equation (2.12)] requires that a be relatively prime to n, but this form does not.

68 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY

2.6 TESTING FOR PRIMALITY

For many cryptographic algorithms, it is necessary to select one or more very large prime numbers at random. Thus, we are faced with the task of determining whether a given large number is prime. There is no simple yet efficient means of accomplish- ing this task.

In this section, we present one attractive and popular algorithm. You may be surprised to learn that this algorithm yields a number that is not necessarily a prime. However, the algorithm can yield a number that is almost certainly a prime. This will be explained presently. We also make reference to a deterministic algorithm for find- ing primes. The section closes with a discussion concerning the distribution of primes.

Miller–Rabin Algorithm9

The algorithm due to Miller and Rabin [MILL75, RABI80] is typically used to test a large number for primality. Before explaining the algorithm, we need some back- ground. First, any positive odd integer n Ú 3 can be expressed as

n – 1 = 2kq with k 7 0, q odd

To see this, note that n – 1 is an even integer. Then, divide (n – 1) by 2 until the result is an odd number q, for a total of k divisions. If n is expressed as a binary number, then the result is achieved by shifting the number to the right until the rightmost digit is a 1, for a total of k shifts. We now develop two properties of prime numbers that we will need.

TWO PROPERTIES OF PRIME NUMBERS The first property is stated as follows: If p is prime and a is a positive integer less than p, then a2 mod p = 1 if and only if either a mod p = 1 or a mod p = -1 mod p = p – 1. By the rules of modular arithmetic (a mod p) (a mod p) = a2 mod p. Thus, if either a mod p = 1 or a mod p = -1, then a2 mod p = 1. Conversely, if a2 mod p = 1, then (a mod p)2 = 1, which is true only for a mod p = 1 or a mod p = -1.

The second property is stated as follows: Let p be a prime number greater than 2. We can then write p – 1 = 2kq with k 7 0, q odd. Let a be any integer in the range 1 6 a 6 p – 1. Then one of the two following conditions is true.

1. aq is congruent to 1 modulo p. That is, aq mod p = 1, or equivalently, aq K 1(mod p).

2. One of the numbers aq, a2q, a4q, c , a2 k – 1q is congruent to -1 mod-

ulo p. That is, there is some number j in the range (1 … j … k) such that a2

j – 1q mod p = -1 mod p = p – 1 or equivalently, a2 j – 1q K – 1(mod p).

Proof: Fermat’s theorem [Equation (2.10)] states that an – 1 K 1(mod n) if n is prime. We have p – 1 = 2kq. Thus, we know that ap – 1 mod p = a2

kq mod p = 1. Thus, if we look at the sequence of numbers

aq mod p, a2q mod p, a4q mod p, c , a2 k – 1q mod p, a2

kq mod p (2.14)

9Also referred to in the literature as the Rabin-Miller algorithm, or the Rabin-Miller test, or the Miller– Rabin test.

2.6 / TESTING FOR PRIMALITY 69

we know that the last number in the list has value 1. Further, each number in the list is the square of the previous number. Therefore, one of the following possibilities must be true.

1. The first number on the list, and therefore all subsequent numbers on the list, equals 1.

2. Some number on the list does not equal 1, but its square mod p does equal 1. By virtue of the first property of prime numbers defined above, we know that the only number that satisfies this condition is p – 1. So, in this case, the list contains an element equal to p – 1.

This completes the proof.

DETAILS OF THE ALGORITHM These considerations lead to the conclusion that, if n is prime, then either the first element in the list of residues, or remainders, (aq, a2q, c , a2

k – 1q, a2 kq) modulo n equals 1; or some element in the list equals

(n – 1); otherwise n is composite (i.e., not a prime). On the other hand, if the condition is met, that does not necessarily mean that n is prime. For example, if n = 2047 = 23 * 89, then n – 1 = 2 * 1023. We compute 21023 mod 2047 = 1, so that 2047 meets the condition but is not prime.

We can use the preceding property to devise a test for primality. The procedure TEST takes a candidate integer n as input and returns the result composite if n is definitely not a prime, and the result inconclusive if n may or may not be a prime.

TEST (n) 1. Find integers k, q, with k > 0, q odd, so that

(n − 1 = 2k q); 2. Select a random integer a, 1 1, (1) If An = Bn, stop. gcd(A, B) = AnCn

(2) If An and Bn are both even, set An + 1 = An/2, Bn + 1 = Bn/2, Cn + 1 = 2Cn

(3) If An is even and Bn is odd, set An + 1 = An/2, Bn + 1 = Bn, Cn + 1 = Cn

(4) If An is odd and Bn is even, set An + 1 = An, Bn + 1 = Bn/2, Cn + 1 = Cn

(5) If An and Bn are both odd, set An + 1 = �An – Bn � , Bn + 1 = min (Bn, An), Cn + 1 = Cn

Continue to step n + 1. a. To get a feel for the two algorithms, compute gcd(6150, 704) using both the Euclid-

ean and Stein’s algorithm. b. What is the apparent advantage of Stein’s algorithm over the Euclidean algorithm?

2.15 a. Show that if Stein’s algorithm does not stop before the nth step, then

Cn + 1 * gcd(An + 1, Bn + 1) = Cn * gcd(An, Bn)

b. Show that if the algorithm does not stop before step (n – 1), then

An + 2Bn + 2 … AnBn

2

c. Show that if 1 … A, B … 2N, then Stein’s algorithm takes at most 4N steps to find gcd(m, n). Thus, Stein’s algorithm works in roughly the same number of steps as the Euclidean algorithm.

d. Demonstrate that Stein’s algorithm does indeed return gcd(A, B). 2.16 Using the extended Euclidean algorithm, find the multiplicative inverse of

a. 135 mod 61 b. 7465 mod 2464 c. 42828 mod 6407

2.17 The purpose of this problem is to determine how many prime numbers there are. Suppose there are a total of n prime numbers, and we list these in order: p1 = 2 6 p2 = 3 6 p3 = 5 6 c 6 pn. a. Define X = 1 + p1p2 c pn. That is, X is equal to one plus the product of all the

primes. Can we find a prime number Pm that divides X? b. What can you say about m? c. Deduce that the total number of primes cannot be finite. d. Show that Pn + 1 … 1 + p1p2 c pn.

2.9 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 81

2.18 The purpose of this problem is to demonstrate that the probability that two random numbers are relatively prime is about 0.6. a. Let P = Pr[gcd(a, b) = 1]. Show that P = Pr[gcd(a, b) = d] = P/d2. Hint:

Consider the quantity gcd aa d

, b d b .

b. The sum of the result of part (a) over all possible values of d is 1. That is Σd Ú1Pr[gcd(a, b) = d] = 1. Use this equality to determine the value of P. Hint:

Use the identity a ∞

i=1

1

i2 = p2

6 .

2.19 Why is gcd(n, n + 1) = 1 for two consecutive integers n and n + 1? 2.20 Using Fermat’s theorem, find 4 2 2 5 mod 13. 2.21 Use Fermat’s theorem to find a number a between 0 and 92 with a congruent to 71013

modulo 93. 2.22 Use Fermat’s theorem to find a number x between 0 and 37 with x 7 3 congruent to 4

modulo 37. (You should not need to use any brute-force searching.) 2.23 Use Euler’s theorem to find a number a between 0 and 9 such that a is congruent to

9 1 0 1 modulo 10. (Note: This is the same as the last digit of the decimal expansion of 9 1 0 0.)

2.24 Use Euler’s theorem to find a number x between 0 and 14 with x 6 1 congruent to 7 modulo 15. (You should not need to use any brute-force searching.)

2.25 Notice in Table 2.6 that f(n) is even for n 7 2. This is true for all n 7 2. Give a con- cise argument why this is so.

2.26 Prove the following: If p is prime, then f(pi) = pi – pi- 1. Hint: What numbers have a factor in common with pi?

2.27 It can be shown (see any book on number theory) that if gcd(m, n) = 1 then f(mn) = f(m)f(n). Using this property, the property developed in the preceding problem, and the property that f(p) = p – 1 for p prime, it is straightforward to determine the value of f(n) for any n. Determine the following: a. f(29) b. f(51) c. f(455) d. f(616)

2.28 It can also be shown that for arbitrary positive integer a, f(a) is given by

f(a) = q t

i=1 [pi

ai – 1(pi – 1)]

where a is given by Equation (2.9), namely: a = P1a1P2a2 c Ptat. Demonstrate this result. 2.29 Consider the function: f(n) = number of elements in the set {a: 0 … a 6 n and

gcd(a, n) = 1}. What is this function? 2.30 Although ancient Chinese mathematicians did good work coming up with their

remainder theorem, they did not always get it right. They had a test for primality. The test said that n is prime if and only if n divides (2n – 2). a. Give an example that satisfies the condition using an odd prime. b. The condition is obviously true for n = 2. Prove that the condition is true if n is an

odd prime (proving the if condition). c. Give an example of an odd n that is not prime and that does not satisfy the condi-

tion. You can do this with nonprime numbers up to a very large value. This misled the Chinese mathematicians into thinking that if the condition is true then n is prime.

d. Unfortunately, the ancient Chinese never tried n = 341, which is nonprime (341 = 11 * 31), yet 341 divides 2341 – 2 without remainder. Demonstrate that 2341 K 2 (mod 341) (disproving the only if condition). Hint: It is not necessary to calculate 2341; play around with the congruences instead.

82 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY

2.31 Show that, if n is an odd composite integer, then the Miller–Rabin test will return inconclusive for a = 1 and a = (n – 1).

2.32 If n is composite and passes the Miller–Rabin test for the base a, then n is called a strong pseudoprime to the base a. Show that 2047 is a strong pseudoprime to the base 2.

2.33 A common formulation of the Chinese remainder theorem (CRT) is as follows: Let m1, c , mk be integers that are pairwise relatively prime for 1 … i, j … k, and i ≠ j. Define M to be the product of all the mi>s. Let a1, c , ak be integers. Then the set of congruences:

x K a1(mod m1) x K a2(mod m2)

~

~

~

x K ak(mod mk)

has a unique solution modulo M. Show that the theorem stated in this form is true. 2.34 The example used by Sun-Tsu to illustrate the CRT was

x K 2 (mod 3); x K 3 (mod 5); x K 2 (mod 7)

Solve for x. 2.35 Six professors begin courses on Monday, Tuesday, Wednesday, Thursday, Friday,

and Saturday, respectively, and announce their intentions of lecturing at intervals of 3, 2, 5, 6, 1, and 4 days, respectively. The regulations of the university forbid Sunday lectures (so that a Sunday lecture must be omitted). When first will all six professors find themselves compelled to omit a lecture? Hint: Use the CRT.

2.36 Find all primitive roots of 37. 2.37 Given 5 as a primitive root of 23, construct a table of discrete logarithms, and use it to

solve the following congruences. a. 3×5 K 2 (mod 23) b. 7×10 + 1 K 0 (mod 23) c. 5x K 6 (mod 23)

Programming Problems

2.1 Write a computer program that implements fast exponentiation (successive squaring) modulo n.

2.2 Write a computer program that implements the Miller–Rabin algorithm for a user- specified n. The program should allow the user two choices: (1) specify a possible witness a to test using the Witness procedure or (2) specify a number s of random witnesses for the Miller–Rabin test to check.

APPENDIX 2A THE MEANING OF MOD

The operator mod is used in this book and in the literature in two different ways: as a binary operator and as a congruence relation. This appendix explains the distinc- tion and precisely defines the notation used in this book regarding parentheses. This notation is common but, unfortunately, not universal.

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APPENDIX 2A / THE MEANING OF MOD 83

The Binary Operator mod

If a is an integer and n is a positive integer, we define a mod n to be the remainder when a is divided by n. The integer n is called the modulus, and the remainder is called the residue. Thus, for any integer a, we can always write

a = :a/n; * n + (a mod n) Formally, we define the operator mod as

a mod n = a – :a/n; * n for n ≠ 0 As a binary operation, mod takes two integer arguments and returns the re-

mainder. For example, 7 mod 3 = 1. The arguments may be integers, integer vari- ables, or integer variable expressions. For example, all of the following are valid, with the obvious meanings:

7 mod 3

7 mod m

x mod 3

x mod m

(x2 + y + 1) mod (2m + n)

where all of the variables are integers. In each case, the left-hand term is divided by the right-hand term, and the resulting value is the remainder. Note that if either the left- or right-hand argument is an expression, the expression is parenthesized. The operator mod is not inside parentheses.

In fact, the mod operation also works if the two arguments are arbitrary real num- bers, not just integers. In this book, we are concerned only with the integer operation.

The Congruence Relation mod

As a congruence relation, mod expresses that two arguments have the same remain- der with respect to a given modulus. For example, 7 K 4 (mod 3) expresses the fact that both 7 and 4 have a remainder of 1 when divided by 3. The following two expressions are equivalent:

a K b (mod m) 3 a mod m = b mod m

Another way of expressing it is to say that the expression a K b (mod m) is the same as saying that a – b is an integral multiple of m. Again, all the arguments may be integers, integer variables, or integer variable expressions. For example, all of the following are valid, with the obvious meanings:

7 K 4 (mod 3) x K y (mod m) (x2 + y + 1) K (a + 1)(mod [m + n])

where all of the variables are integers. Two conventions are used. The congruence sign is K . The modulus for the relation is defined by placing the mod operator fol- lowed by the modulus in parentheses.

84 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY

The congruence relation is used to define residue classes. Those numbers that have the same remainder r when divided by m form a residue class (mod m). There are m residue classes (mod m). For a given remainder r, the residue class to which it belongs consists of the numbers

r, r { m, r { 2m, c

According to our definition, the congruence

a K b (mod m)

signifies that the numbers a and b differ by a multiple of m. Consequently, the con- gruence can also be expressed in the terms that a and b belong to the same residue class (mod m).

85

PART TWO: SYMMETRIC CIPHERS

CHAPTER

Classical Encryption Techniques 3.1 Symmetric Cipher Model

Cryptography Cryptanalysis and Brute-Force Attack

3.2 Substitution Techniques

Caesar Cipher Monoalphabetic Ciphers Playfair Cipher Hill Cipher Polyalphabetic Ciphers One-Time Pad

3.3 Transposition Techniques

3.4 Rotor Machines

3.5 Steganography

3.6 Key Terms, Review Questions, and Problems

86 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES

Symmetric encryption, also referred to as conventional encryption or single-key encryption, was the only type of encryption in use prior to the development of public- key encryption in the 1970s. It remains by far the most widely used of the two types of encryption. Part One examines a number of symmetric ciphers. In this chapter, we begin with a look at a general model for the symmetric encryption process; this will enable us to understand the context within which the algorithms are used. Next, we examine a variety of algorithms in use before the computer era. Finally, we look briefly at a different approach known as steganography. Chapters 4 and 6 introduce the two most widely used symmetric cipher: DES and AES.

Before beginning, we define some terms. An original message is known as the plaintext, while the coded message is called the ciphertext. The process of convert- ing from plaintext to ciphertext is known as enciphering or encryption; restoring the plaintext from the ciphertext is deciphering or decryption. The many schemes used for encryption constitute the area of study known as cryptography. Such a scheme is known as a cryptographic system or a cipher. Techniques used for deciphering a message without any knowledge of the enciphering details fall into the area of crypt- analysis. Cryptanalysis is what the layperson calls “breaking the code.” The areas of cryptography and cryptanalysis together are called cryptology.

3.1 SYMMETRIC CIPHER MODEL

A symmetric encryption scheme has five ingredients (Figure 3.1):

■ Plaintext: This is the original intelligible message or data that is fed into the algorithm as input.

■ Encryption algorithm: The encryption algorithm performs various substitu- tions and transformations on the plaintext.

■ Secret key: The secret key is also input to the encryption algorithm. The key is a value independent of the plaintext and of the algorithm. The algorithm will produce a different output depending on the specific key being used at the time. The exact substitutions and transformations performed by the algorithm depend on the key.

LEARNING OBJECTIVES

After studying this chapter, you should be able to:

◆ Present an overview of the main concepts of symmetric cryptography.

◆ Explain the difference between cryptanalysis and brute-force attack.

◆ Understand the operation of a monoalphabetic substitution cipher.

◆ Understand the operation of a polyalphabetic cipher.

◆ Present an overview of the Hill cipher.

◆ Describe the operation of a rotor machine.

3.1 / SYMMETRIC CIPHER MODEL 87

■ Ciphertext: This is the scrambled message produced as output. It depends on the plaintext and the secret key. For a given message, two different keys will produce two different ciphertexts. The ciphertext is an apparently random stream of data and, as it stands, is unintelligible.

■ Decryption algorithm: This is essentially the encryption algorithm run in reverse. It takes the ciphertext and the secret key and produces the original plaintext.

There are two requirements for secure use of conventional encryption:

1. We need a strong encryption algorithm. At a minimum, we would like the algo- rithm to be such that an opponent who knows the algorithm and has access to one or more ciphertexts would be unable to decipher the ciphertext or figure out the key. This requirement is usually stated in a stronger form: The oppo- nent should be unable to decrypt ciphertext or discover the key even if he or she is in possession of a number of ciphertexts together with the plaintext that produced each ciphertext.

2. Sender and receiver must have obtained copies of the secret key in a secure fashion and must keep the key secure. If someone can discover the key and knows the algorithm, all communication using this key is readable.

We assume that it is impractical to decrypt a message on the basis of the ciphertext plus knowledge of the encryption/decryption algorithm. In other words, we do not need to keep the algorithm secret; we need to keep only the key secret. This feature of symmetric encryption is what makes it feasible for widespread use. The fact that the algorithm need not be kept secret means that manufacturers can and have developed low-cost chip implementations of data encryption algorithms. These chips are widely available and incorporated into a number of products. With the use of symmetric encryption, the principal security problem is maintaining the secrecy of the key.

Let us take a closer look at the essential elements of a symmetric encryp- tion scheme, using Figure 3.2. A source produces a message in plaintext, X = [X1, X2, c , XM]. The M elements of X are letters in some finite alphabet. Traditionally, the alphabet usually consisted of the 26 capital letters. Nowadays,

Figure 3.1 Simplified Model of Symmetric Encryption

Plaintext input

Y = E(K, X ) X = D(K, Y )

X

KK

Transmitted ciphertext

Plaintext output

Secret key shared by sender and recipient

Secret key shared by sender and recipient

Encryption algorithm (e.g., AES)

Decryption algorithm (reverse of encryption

algorithm)

88 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES

the binary alphabet {0, 1} is typically used. For encryption, a key of the form K = [K1, K2, c , KJ] is generated. If the key is generated at the message source, then it must also be provided to the destination by means of some secure channel. Alternatively, a third party could generate the key and securely deliver it to both source and destination.

With the message X and the encryption key K as input, the encryption algo- rithm forms the ciphertext Y = [Y1, Y2, c , YN]. We can write this as

Y = E(K, X)

This notation indicates that Y is produced by using encryption algorithm E as a function of the plaintext X, with the specific function determined by the value of the key K.

The intended receiver, in possession of the key, is able to invert the transformation:

X = D(K, Y)

An opponent, observing Y but not having access to K or X, may attempt to recover X or K or both X and K. It is assumed that the opponent knows the encryp- tion (E) and decryption (D) algorithms. If the opponent is interested in only this particular message, then the focus of the effort is to recover X by generating a plain- text estimate Xn . Often, however, the opponent is interested in being able to read future messages as well, in which case an attempt is made to recover K by generat- ing an estimate Kn .

Figure 3.2 Model of Symmetric Cryptosystem

Message source

Cryptanalyst

Key source

Destination X X

X

K

Y = E(K, X )

Secure channel

K

Encryption algorithm

Decryption algorithm

3.1 / SYMMETRIC CIPHER MODEL 89

Cryptography

Cryptographic systems are characterized along three independent dimensions:

1. The type of operations used for transforming plaintext to ciphertext. All encryption algorithms are based on two general principles: substitution, in which each element in the plaintext (bit, letter, group of bits or letters) is mapped into another element, and transposition, in which elements in the plaintext are rearranged. The fundamental requirement is that no information be lost (i.e., that all operations are reversible). Most systems, referred to as product systems, involve multiple stages of substitutions and transpositions.

2. The number of keys used. If both sender and receiver use the same key, the system is referred to as symmetric, single-key, secret-key, or conventional encryption. If the sender and receiver use different keys, the system is referred to as asymmetric, two-key, or public-key encryption.

3. The way in which the plaintext is processed. A block cipher processes the input one block of elements at a time, producing an output block for each input block. A stream cipher processes the input elements continuously, producing output one element at a time, as it goes along.

Cryptanalysis and Brute-Force Attack

Typically, the objective of attacking an encryption system is to recover the key in use rather than simply to recover the plaintext of a single ciphertext. There are two general approaches to attacking a conventional encryption scheme:

■ Cryptanalysis: Cryptanalytic attacks rely on the nature of the algorithm plus perhaps some knowledge of the general characteristics of the plaintext or even some sample plaintext–ciphertext pairs. This type of attack exploits the charac- teristics of the algorithm to attempt to deduce a specific plaintext or to deduce the key being used.

■ Brute-force attack: The attacker tries every possible key on a piece of cipher- text until an intelligible translation into plaintext is obtained. On average, half of all possible keys must be tried to achieve success.

If either type of attack succeeds in deducing the key, the effect is catastrophic: All future and past messages encrypted with that key are compromised.

We first consider cryptanalysis and then discuss brute-force attacks. Table 3.1 summarizes the various types of cryptanalytic attacks based on the

amount of information known to the cryptanalyst. The most difficult problem is presented when all that is available is the ciphertext only. In some cases, not even the encryption algorithm is known, but in general, we can assume that the opponent does know the algorithm used for encryption. One possible attack under these cir- cumstances is the brute-force approach of trying all possible keys. If the key space is very large, this becomes impractical. Thus, the opponent must rely on an analysis of the ciphertext itself, generally applying various statistical tests to it. To use this

90 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES

approach, the opponent must have some general idea of the type of plaintext that is concealed, such as English or French text, an EXE file, a Java source listing, an accounting file, and so on.

The ciphertext-only attack is the easiest to defend against because the oppo- nent has the least amount of information to work with. In many cases, however, the analyst has more information. The analyst may be able to capture one or more plaintext messages as well as their encryptions. Or the analyst may know that certain plaintext patterns will appear in a message. For example, a file that is encoded in the Postscript format always begins with the same pattern, or there may be a standard- ized header or banner to an electronic funds transfer message, and so on. All these are examples of known plaintext. With this knowledge, the analyst may be able to deduce the key on the basis of the way in which the known plaintext is transformed.

Closely related to the known-plaintext attack is what might be referred to as a probable-word attack. If the opponent is working with the encryption of some gen- eral prose message, he or she may have little knowledge of what is in the message. However, if the opponent is after some very specific information, then parts of the message may be known. For example, if an entire accounting file is being transmit- ted, the opponent may know the placement of certain key words in the header of the file. As another example, the source code for a program developed by Corporation X might include a copyright statement in some standardized position.

If the analyst is able somehow to get the source system to insert into the sys- tem a message chosen by the analyst, then a chosen-plaintext attack is possible. An example of this strategy is differential cryptanalysis, explored in Appendix S.

Type of Attack Known to Cryptanalyst

Ciphertext Only ■ Encryption algorithm ■ Ciphertext

Known Plaintext ■ Encryption algorithm ■ Ciphertext ■ One or more plaintext–ciphertext pairs formed with the secret key

Chosen Plaintext ■ Encryption algorithm ■ Ciphertext ■ Plaintext message chosen by cryptanalyst, together with its corresponding

ciphertext generated with the secret key

Chosen Ciphertext ■ Encryption algorithm ■ Ciphertext ■ Ciphertext chosen by cryptanalyst, together with its corresponding decrypted

plaintext generated with the secret key

Chosen Text ■ Encryption algorithm ■ Ciphertext ■ Plaintext message chosen by cryptanalyst, together with its corresponding

ciphertext generated with the secret key ■ Ciphertext chosen by cryptanalyst, together with its corresponding decrypted

plaintext generated with the secret key

Table 3.1 Types of Attacks on Encrypted Messages

3.1 / SYMMETRIC CIPHER MODEL 91

In general, if the analyst is able to choose the messages to encrypt, the analyst may deliberately pick patterns that can be expected to reveal the structure of the key.

Table 3.1 lists two other types of attack: chosen ciphertext and chosen text. These are less commonly employed as cryptanalytic techniques but are nevertheless possible avenues of attack.

Only relatively weak algorithms fail to withstand a ciphertext-only attack. Generally, an encryption algorithm is designed to withstand a known-plaintext attack.

Two more definitions are worthy of note. An encryption scheme is unconditionally secure if the ciphertext generated by the scheme does not contain enough information to determine uniquely the corresponding plaintext, no matter how much ciphertext is available. That is, no matter how much time an opponent has, it is impossible for him or her to decrypt the ciphertext simply because the required information is not there. With the exception of a scheme known as the one-time pad (described later in this chapter), there is no encryption algorithm that is unconditionally secure. Therefore, all that the users of an encryption algorithm can strive for is an algorithm that meets one or both of the following criteria:

■ The cost of breaking the cipher exceeds the value of the encrypted information.

■ The time required to break the cipher exceeds the useful lifetime of the information.

An encryption scheme is said to be computationally secure if either of the foregoing two criteria are met. Unfortunately, it is very difficult to estimate the amount of effort required to cryptanalyze ciphertext successfully.

All forms of cryptanalysis for symmetric encryption schemes are designed to exploit the fact that traces of structure or pattern in the plaintext may survive encryption and be discernible in the ciphertext. This will become clear as we exam- ine various symmetric encryption schemes in this chapter. We will see in Part Two that cryptanalysis for public-key schemes proceeds from a fundamentally different premise, namely, that the mathematical properties of the pair of keys may make it possible for one of the two keys to be deduced from the other.

A brute-force attack involves trying every possible key until an intelligible translation of the ciphertext into plaintext is obtained. On average, half of all pos- sible keys must be tried to achieve success. That is, if there are X different keys, on average an attacker would discover the actual key after X/2 tries. It is important to note that there is more to a brute-force attack than simply running through all pos- sible keys. Unless known plaintext is provided, the analyst must be able to recognize plaintext as plaintext. If the message is just plain text in English, then the result pops out easily, although the task of recognizing English would have to be automated. If the text message has been compressed before encryption, then recognition is more difficult. And if the message is some more general type of data, such as a numeri- cal file, and this has been compressed, the problem becomes even more difficult to automate. Thus, to supplement the brute-force approach, some degree of knowl- edge about the expected plaintext is needed, and some means of automatically dis- tinguishing plaintext from garble is also needed.

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92 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES

3.2 SUBSTITUTION TECHNIQUES

In this section and the next, we examine a sampling of what might be called classical encryption techniques. A study of these techniques enables us to illustrate the basic approaches to symmetric encryption used today and the types of cryptanalytic at- tacks that must be anticipated.

The two basic building blocks of all encryption techniques are substitution and transposition. We examine these in the next two sections. Finally, we discuss a system that combines both substitution and transposition.

A substitution technique is one in which the letters of plaintext are replaced by other letters or by numbers or symbols.1 If the plaintext is viewed as a sequence of bits, then substitution involves replacing plaintext bit patterns with ciphertext bit patterns.

Caesar Cipher

The earliest known, and the simplest, use of a substitution cipher was by Julius Caesar. The Caesar cipher involves replacing each letter of the alphabet with the letter standing three places further down the alphabet. For example,

plain: meet me after the toga party cipher: PHHW PH DIWHU WKH WRJD SDUWB

Note that the alphabet is wrapped around, so that the letter following Z is A. We can define the transformation by listing all possibilities, as follows:

plain: a b c d e f g h i j k l m n o p q r s t u v w x y z cipher: D E F G H I J K L M N O P Q R S T U V W X Y Z A B C

Let us assign a numerical equivalent to each letter:

a b c d e f g h i j k l m

0 1 2 3 4 5 6 7 8 9 10 11 12

n o p q r s t u v w x y z

13 14 15 16 17 18 19 20 21 22 23 24 25

Then the algorithm can be expressed as follows. For each plaintext letter p, substi- tute the ciphertext letter C:2

C = E(3, p) = (p + 3) mod 26

A shift may be of any amount, so that the general Caesar algorithm is

C = E(k, p) = (p + k) mod 26 (3.1)

1When letters are involved, the following conventions are used in this book. Plaintext is always in lowercase; ciphertext is in uppercase; key values are in italicized lowercase. 2We define a mod n to be the remainder when a is divided by n. For example, 11 mod 7 = 4. See Chapter 2 for a further discussion of modular arithmetic.

3.2 / SUBSTITUTION TECHNIQUES 93

where k takes on a value in the range 1 to 25. The decryption algorithm is simply

p = D(k, C) = (C – k) mod 26 (3.2)

If it is known that a given ciphertext is a Caesar cipher, then a brute-force cryptanalysis is easily performed: simply try all the 25 possible keys. Figure 3.3 shows the results of applying this strategy to the example ciphertext. In this case, the plaintext leaps out as occupying the third line.

Three important characteristics of this problem enabled us to use a brute- force cryptanalysis:

1. The encryption and decryption algorithms are known.

2. There are only 25 keys to try.

3. The language of the plaintext is known and easily recognizable.

In most networking situations, we can assume that the algorithms are known. What generally makes brute-force cryptanalysis impractical is the use of an algo- rithm that employs a large number of keys. For example, the triple DES algorithm,

Figure 3.3 Brute-Force Cryptanalysis of Caesar Cipher

PHHW PH DIWHU WKH WRJD SDUWB KEY

1 oggv og chvgt vjg vqic rctva

2 nffu nf bgufs uif uphb qbsuz

3 meet me after the toga party

4 ldds ld zesdq sgd snfz ozqsx

5 kccr kc ydrcp rfc rmey nyprw

6 jbbq jb xcqbo qeb qldx mxoqv

7 iaap ia wbpan pda pkcw lwnpu

8 hzzo hz vaozm ocz ojbv kvmot

9 gyyn gy uznyl nby niau julns

10 fxxm fx tymxk max mhzt itkmr

11 ewwl ew sxlwj lzw lgys hsjlq

12 dvvk dv rwkvi kyv kfxr grikp

13 cuuj cu qvjuh jxu jewq fqhjo

14 btti bt puitg iwt idvp epgin

15 assh as othsf hvs hcuo dofhm

16 zrrg zr nsgre gur gbtn cnegl

17 yqqf yq mrfqd ftq fasm bmdfk

18 xppe xp lqepc esp ezrl alcej

19 wood wo kpdob dro dyqk zkbdi

20 vnnc vn jocna cqn cxpj yjach

21 ummb um inbmz bpm bwoi xizbg

22 tlla tl hmaly aol avnh whyaf

23 skkz sk glzkx znk zumg vgxze

24 rjjy rj fkyjw ymj ytlf ufwyd

25 qiix qi ejxiv xli xske tevxc

94 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES

examined in Chapter 7, makes use of a 168-bit key, giving a key space of 2168 or greater than 3.7 * 1050 possible keys.

The third characteristic is also significant. If the language of the plaintext is unknown, then plaintext output may not be recognizable. Furthermore, the input may be abbreviated or compressed in some fashion, again making recognition dif- ficult. For example, Figure 3.4 shows a portion of a text file compressed using an algorithm called ZIP. If this file is then encrypted with a simple substitution cipher (expanded to include more than just 26 alphabetic characters), then the plaintext may not be recognized when it is uncovered in the brute-force cryptanalysis.

Monoalphabetic Ciphers

With only 25 possible keys, the Caesar cipher is far from secure. A dramatic increase in the key space can be achieved by allowing an arbitrary substitution. Before pro- ceeding, we define the term permutation. A permutation of a finite set of elements S is an ordered sequence of all the elements of S, with each element appearing exactly once. For example, if S = {a, b, c}, there are six permutations of S:

abc, acb, bac, bca, cab, cba

In general, there are n! permutations of a set of n elements, because the first element can be chosen in one of n ways, the second in n – 1 ways, the third in n – 2 ways, and so on.

Recall the assignment for the Caesar cipher:

plain: a b c d e f g h i j k l m n o p q r s t u v w x y z cipher: D E F G H I J K L M N O P Q R S T U V W X Y Z A B C

If, instead, the “cipher” line can be any permutation of the 26 alphabetic characters, then there are 26! or greater than 4 * 1026 possible keys. This is 10 orders of mag- nitude greater than the key space for DES and would seem to eliminate brute-force techniques for cryptanalysis. Such an approach is referred to as a monoalphabetic substitution cipher, because a single cipher alphabet (mapping from plain alphabet to cipher alphabet) is used per message.

There is, however, another line of attack. If the cryptanalyst knows the nature of the plaintext (e.g., noncompressed English text), then the analyst can exploit the regularities of the language. To see how such a cryptanalysis might proceed, we give a partial example here that is adapted from one in [SINK09]. The ciphertext to be solved is

Figure 3.4 Sample of Compressed Text

3.2 / SUBSTITUTION TECHNIQUES 95

UZQSOVUOHXMOPVGPOZPEVSGZWSZOPFPESXUDBMETSXAIZ VUEPHZHMDZSHZOWSFPAPPDTSVPQUZWYMXUZUHSX EPYEPOPDZSZUFPOMBZWPFUPZHMDJUDTMOHMQ

As a first step, the relative frequency of the letters can be determined and compared to a standard frequency distribution for English, such as is shown in Figure 3.5 (based on [LEWA00]). If the message were long enough, this technique alone might be sufficient, but because this is a relatively short message, we cannot expect an exact match. In any case, the relative frequencies of the letters in the ciphertext (in percentages) are as follows:

P 13.33 H 5.83 F 3.33 B 1.67 C 0.00

Z 11.67 D 5.00 W 3.33 G 1.67 K 0.00

S 8.33 E 5.00 Q 2.50 Y 1.67 L 0.00

U 8.33 V 4.17 T 2.50 I 0.83 N 0.00

O 7.50 X 4.17 A 1.67 J 0.83 R 0.00

M 6.67

Comparing this breakdown with Figure 3.5, it seems likely that cipher letters P and Z are the equivalents of plain letters e and t, but it is not certain which is which. The letters S, U, O, M, and H are all of relatively high frequency and probably

Figure 3.5 Relative Frequency of Letters in English Text

0

2

4

6

8

10

12

14

A

8. 16

7

1. 49

2

2. 78

2

4. 25

3

12 .7

02

2. 22

8

2. 01

5

6. 09

4 6 .9

96

0. 15

3 0. 77

2

4. 02

5

2. 40

6

6. 74

9 7. 50

7

1. 92

9

0. 09

5

5. 98

7

6. 32

7

9. 05

6

2. 75

8

0. 97

8

2. 36

0

0. 15

0

1. 97

4

0. 07

4

B C D E F G H I J K L M N

R el

at iv

e fr

eq ue

nc y

(% )

O P Q R S T U V W X Y Z

96 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES

correspond to plain letters from the set {a, h, i, n, o, r, s}. The letters with the lowest frequencies (namely, A, B, G, Y, I, J) are likely included in the set {b, j, k, q, v, x, z}.

There are a number of ways to proceed at this point. We could make some tentative assignments and start to fill in the plaintext to see if it looks like a rea- sonable “skeleton” of a message. A more systematic approach is to look for other regularities. For example, certain words may be known to be in the text. Or we could look for repeating sequences of cipher letters and try to deduce their plaintext equivalents.

A powerful tool is to look at the frequency of two-letter combinations, known as digrams. A table similar to Figure 3.5 could be drawn up showing the relative fre- quency of digrams. The most common such digram is th. In our ciphertext, the most common digram is ZW, which appears three times. So we make the correspondence of Z with t and W with h. Then, by our earlier hypothesis, we can equate P with e. Now notice that the sequence ZWP appears in the ciphertext, and we can translate that sequence as “the.” This is the most frequent trigram (three-letter combination) in English, which seems to indicate that we are on the right track.

Next, notice the sequence ZWSZ in the first line. We do not know that these four letters form a complete word, but if they do, it is of the form th_t. If so, S equates with a.

So far, then, we have

UZQSOVUOHXMOPVGPOZPEVSGZWSZOPFPESXUDBMETSXAIZ t a e e te a that e e a a VUEPHZHMDZSHZOWSFPAPPDTSVPQUZWYMXUZUHSX

e t ta t ha e ee a e th t a EPYEPOPDZSZUFPOMBZWPFUPZHMDJUDTMOHMQ e e e tat e the t

Only four letters have been identified, but already we have quite a bit of the message. Continued analysis of frequencies plus trial and error should easily yield a solution from this point. The complete plaintext, with spaces added between words, follows:

it was disclosed yesterday that several informal but direct contacts have been made with political representatives of the viet cong in moscow

Monoalphabetic ciphers are easy to break because they reflect the frequency data of the original alphabet. A countermeasure is to provide multiple substi- tutes, known as homophones, for a single letter. For example, the letter e could be assigned a number of different cipher symbols, such as 16, 74, 35, and 21, with each homophone assigned to a letter in rotation or randomly. If the number of symbols assigned to each letter is proportional to the relative frequency of that let- ter, then single-letter frequency information is completely obliterated. The great mathematician Carl Friedrich Gauss believed that he had devised an unbreak- able cipher using homophones. However, even with homophones, each element of plaintext affects only one element of ciphertext, and multiple-letter patterns

3.2 / SUBSTITUTION TECHNIQUES 97

(e.g., digram frequencies) still survive in the ciphertext, making cryptanalysis rela- tively straightforward.

Two principal methods are used in substitution ciphers to lessen the extent to which the structure of the plaintext survives in the ciphertext: One approach is to encrypt multiple letters of plaintext, and the other is to use multiple cipher alpha- bets. We briefly examine each.

Playfair Cipher

The best-known multiple-letter encryption cipher is the Playfair, which treats di- grams in the plaintext as single units and translates these units into ciphertext digrams.3

The Playfair algorithm is based on the use of a 5 * 5 matrix of letters con- structed using a keyword. Here is an example, solved by Lord Peter Wimsey in Dorothy Sayers’s Have His Carcase:4

M O N A R

C H Y B D

E F G I/J K

L P Q S T

U V W X Z

In this case, the keyword is monarchy. The matrix is constructed by filling in the letters of the keyword (minus duplicates) from left to right and from top to bottom, and then filling in the remainder of the matrix with the remaining letters in alphabetic order. The letters I and J count as one letter. Plaintext is encrypted two letters at a time, according to the following rules:

1. Repeating plaintext letters that are in the same pair are separated with a filler letter, such as x, so that balloon would be treated as ba lx lo on.

2. Two plaintext letters that fall in the same row of the matrix are each replaced by the letter to the right, with the first element of the row circularly following the last. For example, ar is encrypted as RM.

3. Two plaintext letters that fall in the same column are each replaced by the let- ter beneath, with the top element of the column circularly following the last. For example, mu is encrypted as CM.

4. Otherwise, each plaintext letter in a pair is replaced by the letter that lies in its own row and the column occupied by the other plaintext letter. Thus, hs becomes BP and ea becomes IM (or JM, as the encipherer wishes).

The Playfair cipher is a great advance over simple monoalphabetic ciphers. For one thing, whereas there are only 26 letters, there are 26 * 26 = 676 digrams,

3This cipher was actually invented by British scientist Sir Charles Wheatstone in 1854, but it bears the name of his friend Baron Playfair of St. Andrews, who championed the cipher at the British foreign office. 4The book provides an absorbing account of a probable-word attack.

98 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES

so that identification of individual digrams is more difficult. Furthermore, the rela- tive frequencies of individual letters exhibit a much greater range than that of digrams, making frequency analysis much more difficult. For these reasons, the Playfair cipher was for a long time considered unbreakable. It was used as the stan- dard field system by the British Army in World War I and still enjoyed considerable use by the U.S. Army and other Allied forces during World War II.

Despite this level of confidence in its security, the Playfair cipher is relatively easy to break, because it still leaves much of the structure of the plaintext language intact. A few hundred letters of ciphertext are generally sufficient.

One way of revealing the effectiveness of the Playfair and other ciphers is shown in Figure 3.6. The line labeled plaintext plots a typical frequency distribution of the 26 alphabetic characters (no distinction between upper and lower case) in ordinary text. This is also the frequency distribution of any monoalphabetic substi- tution cipher, because the frequency values for individual letters are the same, just with different letters substituted for the original letters. The plot is developed in the following way: The number of occurrences of each letter in the text is counted and divided by the number of occurrences of the most frequently used letter. Using the results of Figure 3.5, we see that e is the most frequently used letter. As a result, e has a relative frequency of 1, t of 9.056/12.702 ≈ 0.72, and so on. The points on the horizontal axis correspond to the letters in order of decreasing frequency.

Figure 3.6 also shows the frequency distribution that results when the text is encrypted using the Playfair cipher. To normalize the plot, the number of occur- rences of each letter in the ciphertext was again divided by the number of occur- rences of e in the plaintext. The resulting plot therefore shows the extent to which the frequency distribution of letters, which makes it trivial to solve substitution

Figure 3.6 Relative Frequency of Occurrence of Letters

0 1 2 3 4 5 6 1 7 8 9 10 10 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Plaintext

Playfair

Vigenère

Random polyalphabetic

Frequency ranked letters (decreasing frequency)

N or

m al

iz ed

re la

tiv e

fr eq

ue nc

y

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

3.2 / SUBSTITUTION TECHNIQUES 99

ciphers, is masked by encryption. If the frequency distribution information were totally concealed in the encryption process, the ciphertext plot of frequencies would be flat, and cryptanalysis using ciphertext only would be effectively impossible. As the figure shows, the Playfair cipher has a flatter distribution than does plaintext, but nevertheless, it reveals plenty of structure for a cryptanalyst to work with. The plot also shows the Vigenère cipher, discussed subsequently. The Hill and Vigenère curves on the plot are based on results reported in [SIMM93].

Hill Cipher5

Another interesting multiletter cipher is the Hill cipher, developed by the math- ematician Lester Hill in 1929.

CONCEPTS FROM LINEAR ALGEBRA Before describing the Hill cipher, let us briefly review some terminology from linear algebra. In this discussion, we are concerned with matrix arithmetic modulo 26. For the reader who needs a refresher on matrix multiplication and inversion, see Appendix E.

We define the inverse M-1 of a square matrix M by the equation M(M-1) = M-1M = I, where I is the identity matrix. I is a square matrix that is all zeros except for ones along the main diagonal from upper left to lower right. The inverse of a matrix does not always exist, but when it does, it satisfies the preceding equation. For example,

A = ¢ 5 8 17 3

≤ A-1 mod 26 = ¢9 2 1 15

≤ AA-1 = ¢ (5 * 9) + (8 * 1) (5 * 2) + (8 * 15)

(17 * 9) + (3 * 1) (17 * 2) + (3 * 15) ≤

= ¢ 53 130 156 79

≤ mod 26 = ¢1 0 0 1

≤ To explain how the inverse of a matrix is computed, we begin with the concept

of determinant. For any square matrix (m * m), the determinant equals the sum of all the products that can be formed by taking exactly one element from each row and exactly one element from each column, with certain of the product terms pre- ceded by a minus sign. For a 2 * 2 matrix,

¢k11 k12 k21 k22

≤ the determinant is k11k22 – k12k21. For a 3 * 3 matrix, the value of the determinant is k11k22k33 + k21k32k13 + k31k12k23 – k31k22k13 – k21k12k33 – k11k32k23. If a square

5This cipher is somewhat more difficult to understand than the others in this chapter, but it illustrates an important point about cryptanalysis that will be useful later on. This subsection can be skipped on a first reading.

100 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES

matrix A has a nonzero determinant, then the inverse of the matrix is computed as [A-1]ij = (det A)-1(-1)i+ j(Dji), where (Dji) is the subdeterminant formed by deleting the jth row and the ith column of A, det(A) is the determinant of A, and (det A)-1 is the multiplicative inverse of (det A) mod 26.

Continuing our example,

det ¢ 5 8 17 3

≤ = (5 * 3) – (8 * 17) = -121 mod 26 = 9 We can show that 9-1 mod 26 = 3, because 9 * 3 = 27 mod 26 = 1 (see

Chapter 2 or Appendix E). Therefore, we compute the inverse of A as

A = ¢ 5 8 17 3

≤ A-1 mod 26 = 3¢ 3 -8

-17 5 ≤ = 3¢3 18

9 5 ≤ = ¢ 9 54

27 15 ≤ = ¢9 2

1 15 ≤

THE HILL ALGORITHM This encryption algorithm takes m successive plaintext let- ters and substitutes for them m ciphertext letters. The substitution is determined by m linear equations in which each character is assigned a numerical value (a = 0, b = 1, c , z = 25). For m = 3, the system can be described as

c1 = (k11p1 + k21p2 + k31p3) mod 26

c2 = (k12p1 + k22p2 + k32p3) mod 26

c3 = (k13p1 + k23p2 + k33p3) mod 26

This can be expressed in terms of row vectors and matrices:6

(c1 c2 c3) = (p1 p2 p3)£k11 k12 k13k21 k22 k23 k31 k32 k33

≥ mod 26 or

C = PK mod 26

where C and P are row vectors of length 3 representing the plaintext and ciphertext, and K is a 3 * 3 matrix representing the encryption key. Operations are performed mod 26.

6Some cryptography books express the plaintext and ciphertext as column vectors, so that the column vector is placed after the matrix rather than the row vector placed before the matrix. Sage uses row vec- tors, so we adopt that convention.

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3.2 / SUBSTITUTION TECHNIQUES 101

For example, consider the plaintext “paymoremoney” and use the encryption key

K = £17 17 521 18 21 2 2 19

≥ The first three letters of the plaintext are represented by the vector (15 0 24). Then (15 0 24)K = (303 303 531) mod 26 = (17 17 11) = RRL. Continuing in this fashion, the ciphertext for the entire plaintext is RRLMWBKASPDH.

Decryption requires using the inverse of the matrix K. We can compute det K = 23, and therefore, (det K)-1 mod 26 = 17. We can then compute the inverse as7

K-1 = £ 4 9 1515 17 6 24 0 17

≥ This is demonstrated as

£17 17 521 18 21 2 2 19

≥£ 4 9 1515 17 6 24 0 17

≥ = £443 442 442858 495 780 494 52 365

≥ mod 26 = £1 0 00 1 0 0 0 1

≥ It is easily seen that if the matrix K-1 is applied to the ciphertext, then the

plaintext is recovered. In general terms, the Hill system can be expressed as

C = E(K, P) = PK mod 26

P = D(K, C) = CK-1 mod 26 = PKK-1 = P

As with Playfair, the strength of the Hill cipher is that it completely hides single-letter frequencies. Indeed, with Hill, the use of a larger matrix hides more frequency information. Thus, a 3 * 3 Hill cipher hides not only single-letter but also two-letter frequency information.

Although the Hill cipher is strong against a ciphertext-only attack, it is easily broken with a known plaintext attack. For an m * m Hill cipher, suppose we have m plaintext–ciphertext pairs, each of length m. We label the pairs Pj = (p1jp1j c pmj) and Cj = (c1jc1j c cmj) such that Cj = PjK for 1 … j … m and for some unknown key matrix K. Now define two m * m matrices X = (pij) and Y = (cij). Then we can form the matrix equation Y = XK. If X has an inverse, then we can determine K = X-1Y. If X is not invertible, then a new version of X can be formed with addi- tional plaintext–ciphertext pairs until an invertible X is obtained.

Consider this example. Suppose that the plaintext “hillcipher” is encrypted using a 2 * 2 Hill cipher to yield the ciphertext HCRZSSXNSP. Thus, we know that (7 8)K mod 26 = (7 2); (11 11)K mod 26 = (17 25); and so on. Using the first two plaintext-ciphertext pairs, we have

7The calculations for this example are provided in detail in Appendix E.

102 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES

¢ 7 2 17 25

≤ = ¢ 7 8 11 11

≤K mod 26 The inverse of X can be computed:

¢ 7 8 11 11

≤-1 = ¢25 22 1 23

≤ so

K = ¢25 22 1 23

≤ ¢ 7 2 17 25

≤ = ¢549 600 398 577

≤ mod 26 = ¢3 2 8 5

≤ This result is verified by testing the remaining plaintext–ciphertext pairs.

Polyalphabetic Ciphers

Another way to improve on the simple monoalphabetic technique is to use differ- ent monoalphabetic substitutions as one proceeds through the plaintext message. The general name for this approach is polyalphabetic substitution cipher. All these techniques have the following features in common:

1. A set of related monoalphabetic substitution rules is used.

2. A key determines which particular rule is chosen for a given transformation.

VIGENÈRE CIPHER The best known, and one of the simplest, polyalphabetic ciphers is the Vigenère cipher. In this scheme, the set of related monoalphabetic substitu- tion rules consists of the 26 Caesar ciphers with shifts of 0 through 25. Each cipher is denoted by a key letter, which is the ciphertext letter that substitutes for the plain- text letter a. Thus, a Caesar cipher with a shift of 3 is denoted by the key value 3.8

We can express the Vigenère cipher in the following manner. Assume a sequence of plaintext letters P = p0, p1, p2, c , pn – 1 and a key consisting of the sequence of letters K = k0, k1, k2, c , km – 1, where typically m 6 n. The sequence of ciphertext letters C = C0, C1, C2, c , Cn – 1 is calculated as follows:

C = C0, C1, C2, c , Cn – 1 = E(K, P) = E[(k0, k1, k2, c , km – 1), (p0, p1, p2, c , pn – 1)]

= (p0 + k0) mod 26, (p1 + k1) mod 26, c ,(pm – 1 + km – 1) mod 26,

(pm + k0) mod 26, (pm + 1 + k1) mod 26, c , (p2m – 1 + km – 1) mod 26, c

Thus, the first letter of the key is added to the first letter of the plaintext, mod 26, the second letters are added, and so on through the first m letters of the plaintext. For the next m letters of the plaintext, the key letters are repeated. This process

8To aid in understanding this scheme and also to aid in it use, a matrix known as the Vigenère tableau is often used. This tableau is discussed in a document at box.com/Crypto7e.

3.2 / SUBSTITUTION TECHNIQUES 103

continues until all of the plaintext sequence is encrypted. A general equation of the encryption process is

Ci = (pi + ki mod m) mod 26 (3.3)

Compare this with Equation (3.1) for the Caesar cipher. In essence, each plain- text character is encrypted with a different Caesar cipher, depending on the corre- sponding key character. Similarly, decryption is a generalization of Equation (3.2):

pi = (Ci – ki mod m) mod 26 (3.4)

To encrypt a message, a key is needed that is as long as the message. Usually, the key is a repeating keyword. For example, if the keyword is deceptive, the mes- sage “we are discovered save yourself” is encrypted as

key: deceptivedeceptivedeceptive plaintext: wearediscoveredsaveyourself ciphertext: ZICVTWQNGRZGVTWAVZHCQYGLMGJ

Expressed numerically, we have the following result.

key 3 4 2 4 15 19 8 21 4 3 4 2 4 15

plaintext 22 4 0 17 4 3 8 18 2 14 21 4 17 4

ciphertext 25 8 2 21 19 22 16 13 6 17 25 6 21 19

key 19 8 21 4 3 4 2 4 15 19 8 21 4

plaintext 3 18 0 21 4 24 14 20 17 18 4 11 5

ciphertext 22 0 21 25 7 2 16 24 6 11 12 6 9

The strength of this cipher is that there are multiple ciphertext letters for each plaintext letter, one for each unique letter of the keyword. Thus, the letter fre- quency information is obscured. However, not all knowledge of the plaintext struc- ture is lost. For example, Figure 3.6 shows the frequency distribution for a Vigenère cipher with a keyword of length 9. An improvement is achieved over the Playfair cipher, but considerable frequency information remains.

It is instructive to sketch a method of breaking this cipher, because the method reveals some of the mathematical principles that apply in cryptanalysis.

First, suppose that the opponent believes that the ciphertext was encrypted using either monoalphabetic substitution or a Vigenère cipher. A simple test can be made to make a determination. If a monoalphabetic substitution is used, then the statistical properties of the ciphertext should be the same as that of the lan- guage of the plaintext. Thus, referring to Figure 3.5, there should be one cipher let- ter with a relative frequency of occurrence of about 12.7%, one with about 9.06%, and so on. If only a single message is available for analysis, we would not expect an exact match of this small sample with the statistical profile of the plaintext lan- guage. Nevertheless, if the correspondence is close, we can assume a monoalpha- betic substitution.

104 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES

If, on the other hand, a Vigenère cipher is suspected, then progress depends on determining the length of the keyword, as will be seen in a moment. For now, let us concentrate on how the keyword length can be determined. The important insight that leads to a solution is the following: If two identical sequences of plaintext let- ters occur at a distance that is an integer multiple of the keyword length, they will generate identical ciphertext sequences. In the foregoing example, two instances of the sequence “red” are separated by nine character positions. Consequently, in both cases, r is encrypted using key letter e, e is encrypted using key letter p, and d is encrypted using key letter t. Thus, in both cases, the ciphertext sequence is VTW. We indicate this above by underlining the relevant ciphertext letters and shading the relevant ciphertext numbers.

An analyst looking at only the ciphertext would detect the repeated sequences VTW at a displacement of 9 and make the assumption that the keyword is either three or nine letters in length. The appearance of VTW twice could be by chance and may not reflect identical plaintext letters encrypted with identical key letters. However, if the message is long enough, there will be a number of such repeated ciphertext sequences. By looking for common factors in the displacements of the vari- ous sequences, the analyst should be able to make a good guess of the keyword length.

Solution of the cipher now depends on an important insight. If the keyword length is m, then the cipher, in effect, consists of m monoalphabetic substitution ciphers. For example, with the keyword DECEPTIVE, the letters in positions 1, 10, 19, and so on are all encrypted with the same monoalphabetic cipher. Thus, we can use the known frequency characteristics of the plaintext language to attack each of the monoalphabetic ciphers separately.

The periodic nature of the keyword can be eliminated by using a nonrepeating keyword that is as long as the message itself. Vigenère proposed what is referred to as an autokey system, in which a keyword is concatenated with the plaintext itself to provide a running key. For our example,

key: deceptivewearediscoveredsav plaintext: wearediscoveredsaveyourself ciphertext: ZICVTWQNGKZEIIGASXSTSLVVWLA

Even this scheme is vulnerable to cryptanalysis. Because the key and the plaintext share the same frequency distribution of letters, a statistical technique can be applied. For example, e enciphered by e, by Figure 3.5, can be expected to occur with a frequency of (0.127)2 ≈ 0.016, whereas t enciphered by t would occur only about half as often. These regularities can be exploited to achieve successful cryptanalysis.9

VERNAM CIPHER The ultimate defense against such a cryptanalysis is to choose a keyword that is as long as the plaintext and has no statistical relationship to it. Such a system was introduced by an AT&T engineer named Gilbert Vernam in 1918.

9Although the techniques for breaking a Vigenère cipher are by no means complex, a 1917 issue of Scientific American characterized this system as “impossible of translation.” This is a point worth remem- bering when similar claims are made for modern algorithms.

3.2 / SUBSTITUTION TECHNIQUES 105

His system works on binary data (bits) rather than letters. The system can be expressed succinctly as follows (Figure 3.7):

ci = pi⊕ ki

where

pi = ith binary digit of plaintext ki = ith binary digit of key ci = ith binary digit of ciphertext ⊕ = exclusive@or (XOR) operation

Compare this with Equation (3.3) for the Vigenère cipher. Thus, the ciphertext is generated by performing the bitwise XOR of the plain-

text and the key. Because of the properties of the XOR, decryption simply involves the same bitwise operation:

pi = ci⊕ ki

which compares with Equation (3.4). The essence of this technique is the means of construction of the key. Vernam

proposed the use of a running loop of tape that eventually repeated the key, so that in fact the system worked with a very long but repeating keyword. Although such a scheme, with a long key, presents formidable cryptanalytic difficulties, it can be broken with sufficient ciphertext, the use of known or probable plaintext sequences, or both.

One-Time Pad

An Army Signal Corp officer, Joseph Mauborgne, proposed an improvement to the Vernam cipher that yields the ultimate in security. Mauborgne suggested using a random key that is as long as the message, so that the key need not be repeated. In addition, the key is to be used to encrypt and decrypt a single message, and then is discarded. Each new message requires a new key of the same length as the new mes- sage. Such a scheme, known as a one-time pad, is unbreakable. It produces random output that bears no statistical relationship to the plaintext. Because the ciphertext

Figure 3.7 Vernam Cipher

Key stream generator

Cryptographic bit stream (ki)

Cryptographic bit stream (ki)

Plaintext (pi)

Plaintext (pi)

Ciphertext (ci )

Key stream generator

106 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES

contains no information whatsoever about the plaintext, there is simply no way to break the code.

An example should illustrate our point. Suppose that we are using a Vigenère scheme with 27 characters in which the twenty-seventh character is the space character, but with a one-time key that is as long as the message. Consider the ciphertext

ANKYODKYUREPFJBYOJDSPLREYIUNOFDOIUERFPLUYTS

We now show two different decryptions using two different keys:

ciphertext: ANKYODKYUREPFJBYOJDSPLREYIUNOFDOIUERFPLUYTS key: pxlmvmsydofuyrvzwc tnlebnecvgdupahfzzlmnyih plaintext: mr mustard with the candlestick in the hall

ciphertext: ANKYODKYUREPFJBYOJDSPLREYIUNOFDOIUERFPLUYTS key: pftgpmiydgaxgoufhklllmhsqdqogtewbqfgyovuhwt plaintext: miss scarlet with the knife in the library

Suppose that a cryptanalyst had managed to find these two keys. Two plau- sible plaintexts are produced. How is the cryptanalyst to decide which is the correct decryption (i.e., which is the correct key)? If the actual key were produced in a truly random fashion, then the cryptanalyst cannot say that one of these two keys is more likely than the other. Thus, there is no way to decide which key is correct and there- fore which plaintext is correct.

In fact, given any plaintext of equal length to the ciphertext, there is a key that produces that plaintext. Therefore, if you did an exhaustive search of all possible keys, you would end up with many legible plaintexts, with no way of knowing which was the intended plaintext. Therefore, the code is unbreakable.

The security of the one-time pad is entirely due to the randomness of the key. If the stream of characters that constitute the key is truly random, then the stream of characters that constitute the ciphertext will be truly random. Thus, there are no patterns or regularities that a cryptanalyst can use to attack the ciphertext.

In theory, we need look no further for a cipher. The one-time pad offers com- plete security but, in practice, has two fundamental difficulties:

1. There is the practical problem of making large quantities of random keys. Any heavily used system might require millions of random characters on a regular basis. Supplying truly random characters in this volume is a significant task.

2. Even more daunting is the problem of key distribution and protection. For every message to be sent, a key of equal length is needed by both sender and receiver. Thus, a mammoth key distribution problem exists.

Because of these difficulties, the one-time pad is of limited utility and is useful primarily for low-bandwidth channels requiring very high security.

The one-time pad is the only cryptosystem that exhibits what is referred to as perfect secrecy. This concept is explored in Appendix F.

3.3 / TRANSPOSITION TECHNIQUES 107

3.3 TRANSPOSITION TECHNIQUES

All the techniques examined so far involve the substitution of a ciphertext symbol for a plaintext symbol. A very different kind of mapping is achieved by performing some sort of permutation on the plaintext letters. This technique is referred to as a transposition cipher.

The simplest such cipher is the rail fence technique, in which the plaintext is written down as a sequence of diagonals and then read off as a sequence of rows. For example, to encipher the message “meet me after the toga party” with a rail fence of depth 2, we write the following:

m e m a t r h t g p r y e t e f e t e o a a t

The encrypted message is

MEMATRHTGPRYETEFETEOAAT

This sort of thing would be trivial to cryptanalyze. A more complex scheme is to write the message in a rectangle, row by row, and read the message off, column by column, but permute the order of the columns. The order of the columns then becomes the key to the algorithm. For example,

Key: 4 3 1 2 5 6 7 Plaintext: a t t a c k p o s t p o n e d u n t i l t w o a m x y z Ciphertext: TTNAAPTMTSUOAODWCOIXKNLYPETZ

Thus, in this example, the key is 4312567. To encrypt, start with the column that is labeled 1, in this case column 3. Write down all the letters in that column. Proceed to column 4, which is labeled 2, then column 2, then column 1, then columns 5, 6, and 7.

A pure transposition cipher is easily recognized because it has the same letter frequencies as the original plaintext. For the type of columnar transposition just shown, cryptanalysis is fairly straightforward and involves laying out the cipher- text in a matrix and playing around with column positions. Digram and trigram fre- quency tables can be useful.

The transposition cipher can be made significantly more secure by perform- ing more than one stage of transposition. The result is a more complex permutation that is not easily reconstructed. Thus, if the foregoing message is reencrypted using the same algorithm,

108 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES

Key: 4 3 1 2 5 6 7 Input: t t n a a p t m t s u o a o d w c o i x k n l y p e t z Output: NSCYAUOPTTWLTMDNAOIEPAXTTOKZ

To visualize the result of this double transposition, designate the letters in the original plaintext message by the numbers designating their position. Thus, with 28 letters in the message, the original sequence of letters is

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

After the first transposition, we have

03 10 17 24 04 11 18 25 02 09 16 23 01 08 15 22 05 12 19 26 06 13 20 27 07 14 21 28

which has a somewhat regular structure. But after the second transposition, we have

17 09 05 27 24 16 12 07 10 02 22 20 03 25 15 13 04 23 19 14 11 01 26 21 18 08 06 28

This is a much less structured permutation and is much more difficult to cryptanalyze.

3.4 ROTOR MACHINES

The example just given suggests that multiple stages of encryption can produce an algorithm that is significantly more difficult to cryptanalyze. This is as true of substi- tution ciphers as it is of transposition ciphers. Before the introduction of DES, the most important application of the principle of multiple stages of encryption was a class of systems known as rotor machines.10

The basic principle of the rotor machine is illustrated in Figure 3.8. The machine consists of a set of independently rotating cylinders through which electri- cal pulses can flow. Each cylinder has 26 input pins and 26 output pins, with internal wiring that connects each input pin to a unique output pin. For simplicity, only three of the internal connections in each cylinder are shown.

If we associate each input and output pin with a letter of the alphabet, then a single cylinder defines a monoalphabetic substitution. For example, in Figure 3.8, if an operator depresses the key for the letter A, an electric signal is applied to

10Machines based on the rotor principle were used by both Germany (Enigma) and Japan (Purple) in World War II. The breaking of both codes by the Allies was a significant factor in the war’s outcome.

3.4 / ROTOR MACHINES 109

the first pin of the first cylinder and flows through the internal connection to the twenty-fifth output pin.

Consider a machine with a single cylinder. After each input key is depressed, the cylinder rotates one position, so that the internal connections are shifted accord- ingly. Thus, a different monoalphabetic substitution cipher is defined. After 26 let- ters of plaintext, the cylinder would be back to the initial position. Thus, we have a polyalphabetic substitution algorithm with a period of 26.

A single-cylinder system is trivial and does not present a formidable crypt- analytic task. The power of the rotor machine is in the use of multiple cylinders, in which the output pins of one cylinder are connected to the input pins of the next. Figure 3.8 shows a three-cylinder system. The left half of the figure shows a position in which the input from the operator to the first pin (plaintext letter a) is routed through the three cylinders to appear at the output of the second pin (ciphertext letter B).

With multiple cylinders, the one closest to the operator input rotates one pin position with each keystroke. The right half of Figure 3.8 shows the system’s configuration after a single keystroke. For every complete rotation of the inner cylinder, the middle cylinder rotates one pin position. Finally, for every complete rotation of the middle cylinder, the outer cylinder rotates one pin position. This is the same type of operation seen with an odometer. The result is that there are 26 * 26 * 26 = 17,576 different substitution alphabets used before the system

Figure 3.8 Three-Rotor Machine with Wiring Represented by Numbered Contacts

24 25 26 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

21 3

15 1

19 10 14 26 20 8

16 7

22 4

11 5

17 9

12 23 18 2

25 6

24 13

A B C D E F G H I J

K L M N O P Q R S T U V W X Y Z

26 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

20 1 6 4

15 3

14 12 23 5

16 2

22 19 11 18 25 24 13 7

10 8

21 9

26 17

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

8 18 26 17 20 22 10 3

13 11 4

23 5

24 9

12 25 16 19 6

15 21 2 7 1

14

A B C D E F G H I J

K L M N O P Q R S T U V W X Y Z

Direction of motion Direction of motion

Fast rotor Medium rotor Slow rotor Fast rotor Medium rotor Slow rotor (a) Initial setting (b) Setting after one keystroke

A B C D E F G H I J

K L M N O P Q R S T U V W X Y Z

23 24 25 26 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

13 21 3

15 1

19 10 14 26 20 8

16 7

22 4

11 5

17 9

12 23 18 2

25 6

24

26 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

20 1 6 4

15 3

14 12 23 5

16 2

22 19 11 18 25 24 13 7

10 8

21 9

26 17

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

8 18 26 17 20 22 10 3

13 11 4

23 5

24 9

12 25 16 19 6

15 21 2 7 1

14

A B C D E F G H I J

K L M N O P Q R S T U V W X Y Z

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110 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES

repeats. The addition of fourth and fifth rotors results in periods of 456,976 and 11,881,376 letters, respectively. Thus, a given setting of a 5-rotor machine is equiva- lent to a Vigenère cipher with a key length of 11,881,376.

Such a scheme presents a formidable cryptanalytic challenge. If, for example, the cryptanalyst attempts to use a letter frequency analysis approach, the analyst is faced with the equivalent of over 11 million monoalphabetic ciphers. We might need on the order of 50 letters in each monalphabetic cipher for a solution, which means that the analyst would need to be in possession of a ciphertext with a length of over half a billion letters.

The significance of the rotor machine today is that it points the way to a large class of symmetric ciphers, of which the Data Encryption Standard (DES) is the most prominent. DES is introduced in Chapter 4.

3.5 STEGANOGRAPHY

We conclude with a discussion of a technique that (strictly speaking), is not encryp- tion, namely, steganography.

A plaintext message may be hidden in one of two ways. The methods of steganography conceal the existence of the message, whereas the methods of cryp- tography render the message unintelligible to outsiders by various transformations of the text.11

A simple form of steganography, but one that is time-consuming to construct, is one in which an arrangement of words or letters within an apparently innocuous text spells out the real message. For example, the sequence of first letters of each word of the overall message spells out the hidden message. Figure 3.9 shows an example in which a subset of the words of the overall message is used to convey the hidden message. See if you can decipher this; it’s not too hard.

Various other techniques have been used historically; some examples are the following [MYER91]:

■ Character marking: Selected letters of printed or typewritten text are over- written in pencil. The marks are ordinarily not visible unless the paper is held at an angle to bright light.

■ Invisible ink: A number of substances can be used for writing but leave no vis- ible trace until heat or some chemical is applied to the paper.

■ Pin punctures: Small pin punctures on selected letters are ordinarily not vis- ible unless the paper is held up in front of a light.

■ Typewriter correction ribbon: Used between lines typed with a black ribbon, the results of typing with the correction tape are visible only under a strong light.

11Steganography was an obsolete word that was revived by David Kahn and given the meaning it has today [KAHN96].

3.5 / STEGANOGRAPHY 111

Although these techniques may seem archaic, they have contemporary equiv- alents. [WAYN09] proposes hiding a message by using the least significant bits of frames on a CD. For example, the Kodak Photo CD format’s maximum resolution is 3096 * 6144 pixels, with each pixel containing 24 bits of RGB color information. The least significant bit of each 24-bit pixel can be changed without greatly affecting the quality of the image. The result is that you can hide a 130-kB message in a single digital snapshot. There are now a number of software packages available that take this type of approach to steganography.

Steganography has a number of drawbacks when compared to encryption. It requires a lot of overhead to hide a relatively few bits of information, although using a scheme like that proposed in the preceding paragraph may make it more effective. Also, once the system is discovered, it becomes virtually worthless. This problem, too, can be overcome if the insertion method depends on some sort of key (e.g., see Problem 3.22). Alternatively, a message can be first encrypted and then hidden using steganography.

The advantage of steganography is that it can be employed by parties who have something to lose should the fact of their secret communication (not necessar- ily the content) be discovered. Encryption flags traffic as important or secret or may identify the sender or receiver as someone with something to hide.

Figure 3.9 A Puzzle for Inspector Morse

(From The Silent World of Nicholas Quinn, by Colin Dexter)

112 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES

3.6 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS

Key Terms

block cipher brute-force attack Caesar cipher cipher ciphertext computationally secure conventional encryption cryptanalysis cryptographic system cryptography

cryptology deciphering decryption digram enciphering encryption Hill cipher monoalphabetic cipher one-time pad plaintext

Playfair cipher polyalphabetic cipher rail fence cipher single-key encryption steganography stream cipher symmetric encryption transposition cipher unconditionally secure Vigenère cipher

Review Questions

3.1 Describe the main requirements for the secure use of symmetric encryption. 3.2 What are the two basic functions used in encryption algorithms? 3.3 Differentiate between secret-key encryption and public-key encryption. 3.4 What is the difference between a block cipher and a stream cipher? 3.5 What are the two general approaches to attacking a cipher? 3.6 List and briefly define types of cryptanalytic attacks based on what is known to the

attacker. 3.7 What is the difference between an unconditionally secure cipher and a computation-

ally secure cipher? 3.8 Why is the Caesar cipher substitution technique vulnerable to a brute-force cryptanalysis? 3.9 How much key space is available when a monoalphabetic substitution cipher is used

to replace plaintext with ciphertext? 3.10 What is the drawback of a Playfair cipher? 3.11 What is the difference between a monoalphabetic cipher and a polyalphabetic cipher? 3.12 What are two problems with the one-time pad? 3.13 What is a transposition cipher? 3.14 What are the drawbacks of Steganography?

Problems

3.1 A generalization of the Caesar cipher, known as the affine Caesar cipher, has the fol- lowing form: For each plaintext letter p, substitute the ciphertext letter C:

C = E([a, b], p) = (ap + b) mod 26

A basic requirement of any encryption algorithm is that it be one-to-one. That is, if p ≠ q, then E(k, p) ≠ E(k, q). Otherwise, decryption is impossible, because more than one plaintext character maps into the same ciphertext character. The affine Caesar cipher is not one-to-one for all values of a. For example, for a = 2 and b = 3, then E([a, b], 0) = E([a, b], 13) = 3.

a. Are there any limitations on the value of b? Explain why or why not. b. Determine which values of a are not allowed.

3.6 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 113

c. Provide a general statement of which values of a are and are not allowed. Justify your statement.

3.2 How many one-to-one affine Caesar ciphers are there? 3.3 A ciphertext has been generated with an affine cipher. The most frequent letter of

the ciphertext is “C,” and the second most frequent letter of the ciphertext is “Z.” Break this code.

3.4 The following ciphertext was generated using a simple substitution algorithm.

hzsrnqc klyy wqc flo mflwf ol zqdn nsoznj wskn lj xzsrbjnf, wzsxz gqv zqhhnf ol ozn glco zlfnco hnlhrn; nsoznj jnrqosdnc lj fnqj kjsnfbc, wzsxz sc xnjoqsfrv gljn efeceqr. zn rsdnb qrlfn sf zsc zlecn sf cqdsrrn jlw, wzsoznj flfn hnfnojqonb. q csfyrn blgncosx cekksxnb ol cnjdn zsg. zn pjnqmkqconb qfb bsfnb qo ozn xrep, qo zlejc gqozngqosxqrrv ksanb, sf ozn cqgn jllg, qo ozn cqgn oqprn, fndnj oqmsfy zsc gnqrc wsoz loznj gngpnjc, gexz rncc pjsfysfy q yenco wsoz zsg; qfb wnfo zlgn qo naqxorv gsbfsyzo, lfrv ol jnosjn qo lfxn ol pnb. zn fndnj ecnb ozn xlcv xzqgpnjc wzsxz ozn jnkljg hjldsbnc klj soc kqdlejnb gngpnjc. zn hqccnb onf zlejc leo lk ozn ownfov-klej sf cqdsrrn jlw, nsoznj sf crnnhsfy lj gqmsfy zsc olsrno.

Decrypt this message. Hints:

1. As you know, the most frequently occurring letter in English is e. Therefore, the first or second (or perhaps third?) most common character in the message is likely to stand for e. Also, e is often seen in pairs (e.g., meet, fleet, speed, seen, been, agree, etc.). Try to find a character in the ciphertext that decodes to e.

2. The most common word in English is “the.” Use this fact to guess the characters that stand for t and h.

3. Decipher the rest of the message by deducing additional words. Warning: The resulting message is in English but may not make much sense on a first

reading. 3.5 One way to solve the key distribution problem is to use a line from a book that both

the sender and the receiver possess. Typically, at least in spy novels, the first sentence of a book serves as the key. The particular scheme discussed in this problem is from one of the best suspense novels involving secret codes, Talking to Strange Men, by Ruth Rendell. Work this problem without consulting that book!

Consider the following message:

SIDKHKDM AF HCRKIABIE SHIMC KD LFEAILA

This ciphertext was produced using the first sentence of The Other Side of Silence (a book about the spy Kim Philby):

The snow lay thick on the steps and the snowflakes driven by the wind looked black in the headlights of the cars.

A simple substitution cipher was used. a. What is the encryption algorithm? b. How secure is it? c. To make the key distribution problem simple, both parties can agree to use the first or

last sentence of a book as the key. To change the key, they simply need to agree on a new book. The use of the first sentence would be preferable to the use of the last. Why?

3.6 In one of his cases, Sherlock Holmes was confronted with the following message.

534 C2 13 127 36 31 4 17 21 41 DOUGLAS 109 293 5 37 BIRLSTONE

26 BIRLSTONE 9 127 171

114 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES

Although Watson was puzzled, Holmes was able immediately to deduce the type of cipher. Can you?

3.7 This problem uses a real-world example, from an old U.S. Special Forces manual (public domain). The document, filename SpecialForces.pdf, is available at box.com/ Crypto7e. a. Using the two keys (memory words) cryptographic and network security, encrypt

the following message:

Be at the third pillar from the left outside the lyceum theatre tonight at seven. If you are distrustful bring two friends.

Make reasonable assumptions about how to treat redundant letters and excess letters in the memory words and how to treat spaces and punctuation. Indicate what your assumptions are. Note: The message is from the Sherlock Holmes novel, The Sign of Four.

b. Decrypt the ciphertext. Show your work. c. Comment on when it would be appropriate to use this technique and what its

advantages are.

3.8 A disadvantage of the general monoalphabetic cipher is that both sender and receiver must commit the permuted cipher sequence to memory. A common technique for avoiding this is to use a keyword from which the cipher sequence can be gener- ated. For example, using the keyword CRYPTO, write out the keyword followed by unused letters in normal order and match this against the plaintext letters:

plain: a b c d e f g h i j k l m n o p q r s t u v w x y z

cipher: C R Y P T O A B D E F G H I J K L M N Q S U V W X Z

If it is felt that this process does not produce sufficient mixing, write the remain- ing letters on successive lines and then generate the sequence by reading down the columns:

C R Y P T O

A B D E F G

H I J K L M

N Q S U V W

X Z

This yields the sequence:

C A H N X R B I Q Z Y D J S P E K U T F L V O G M W

Such a system is used in the example in Section 3.2 (the one that begins “it was disclosed yesterday”). Determine the keyword.

3.9 When the PT-109 American patrol boat, under the command of Lieutenant John F. Kennedy, was sunk by a Japanese destroyer, a message was received at an Australian wireless station in Playfair code:

KXJEY UREBE ZWEHE WRYTU HEYFS

KREHE GOYFI WTTTU OLKSY CAJPO

BOTEI ZONTX BYBNT GONEY CUZWR

GDSON SXBOU YWRHE BAAHY USEDQ

The key used was royal new zealand navy. Decrypt the message. Translate TT into tt.

3.6 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 115

3.10 a. Construct a Playfair matrix with the key algorithm. b. Construct a Playfair matrix with the key cryptography. Make a reasonable assump-

tion about how to treat redundant letters in the key. 3.11 a. Using this Playfair matrix:

J/K C D E F

U N P Q S

Z V W X Y

R A L G O

B I T H M

Encrypt this message:

I only regret that I have but one life to give for my country.

Note: This message is by Nathan Hale, a soldier in the American Revolutionary War. b. Repeat part (a) using the Playfair matrix from Problem 3.10a. c. How do you account for the results of this problem? Can you generalize your

conclusion? 3.12 a. How many possible keys does the Playfair cipher have? Ignore the fact that

some keys might produce identical encryption results. Express your answer as an approximate power of 2.

b. Now take into account the fact that some Playfair keys produce the same encryp- tion results. How many effectively unique keys does the Playfair cipher have?

3.13 What substitution system results when we use a 1 * 25 Playfair matrix? 3.14 a. Encrypt the message “meet me at the usual place at ten rather than eight o clock”

using the Hill cipher with the key ¢7 3 2 5

≤. Show your calculations and the result. b. Show the calculations for the corresponding decryption of the ciphertext to

recover the original plaintext. 3.15 We have shown that the Hill cipher succumbs to a known plaintext attack if sufficient

plaintext–ciphertext pairs are provided. It is even easier to solve the Hill cipher if a chosen plaintext attack can be mounted. Describe such an attack.

3.16 It can be shown that the Hill cipher with the matrix ¢a b c d

≤ requires that (ad – bc) is relatively prime to 26; that is, the only common positive integer factor of (ad – bc) and 26 is 1. Thus, if (ad – bc) = 13 or is even, the matrix is not allowed. Determine the number of different (good) keys there are for a 2 * 2 Hill cipher without count- ing them one by one, using the following steps: a. Find the number of matrices whose determinant is even because one or both rows

are even. (A row is “even” if both entries in the row are even.) b. Find the number of matrices whose determinant is even because one or both col-

umns are even. (A column is “even” if both entries in the column are even.) c. Find the number of matrices whose determinant is even because all of the entries

are odd. d. Taking into account overlaps, find the total number of matrices whose determi-

nant is even. e. Find the number of matrices whose determinant is a multiple of 13 because the

first column is a multiple of 13.

116 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES

f. Find the number of matrices whose determinant is a multiple of 13 where the first column is not a multiple of 13 but the second column is a mul- tiple of the first modulo 13.

g. Find the total number of matrices whose determinant is a multiple of 13. h. Find the number of matrices whose determinant is a multiple of 26

because they fit cases parts (a) and (e), (b) and (e), (c) and (e), (a) and (f), and so on.

i. Find the total number of matrices whose determinant is neither a mul- tiple of 2 nor a multiple of 13.

3.17 Calculate the determinant mod 26 of

a. ¢2 3 5 1 3 7

≤ b. £2 1 1 3 2 55 7 1 8 3 1 4 1 2

≥ 3.18 Determine the inverse mod 26 of

a. ¢2 3 1 22

≤ b. £ 6 24 113 16 10 20 17 15

≥ 3.19 Using the Vigenère cipher, encrypt the word “cryptographic” using the word

“eng”. 3.20 This problem explores the use of a one-time pad version of the Vigenère

cipher. In this scheme, the key is a stream of random numbers between 0 and 26. For example, if the key is 3 19 5 . . . , then the first letter of plaintext is encrypted with a shift of 3 letters, the second with a shift of 19 letters, the third with a shift of 5 letters, and so on. a. Encrypt the plaintext sendmoremoney with the key stream

3 11 5 7 17 21 0 11 14 8 7 13 9

b. Using the ciphertext produced in part (a), find a key so that the cipher- text decrypts to the plaintext cashnotneeded.

3.21 What is the message embedded in Figure 3.9? 3.22 In one of Dorothy Sayers’s mysteries, Lord Peter is confronted with the

message shown in Figure 3.10. He also discovers the key to the message, which is a sequence of integers:

787656543432112343456567878878765654

3432112343456567878878765654433211234

a. Decrypt the message. Hint: What is the largest integer value? b. If the algorithm is known but not the key, how secure is the scheme? c. If the key is known but not the algorithm, how secure is the scheme?

Figure 3.10 A Puzzle for Lord Peter

I thought to see the fairies in the fields, but I saw only the evil elephants with their black backs. Woe! how that sight awed me! The elves danced all around and about while I heard voices calling clearly. Ah! how I tried to see—throw off the ugly cloud—but no blind eye of a mortal was permitted to spy them. So then came minstrels, having gold trumpets, harps and drums. These played very loudly beside me, breaking that spell. So the dream vanished, whereat I thanked Heaven. I shed many tears before the thin moon rose up, frail and faint as a sickle of straw. Now though the Enchanter gnash his teeth vainly, yet shall he return as the Spring returns. Oh, wretched man! Hell gapes, Erebus now lies open. The mouths of Death wait on thy end.

3.6 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 117

Programming Problems

3.23 Write a program that can encrypt and decrypt using the general Caesar cipher, also known as an additive cipher.

3.24 Write a program that can encrypt and decrypt using the affine cipher described in Problem 3.1.

3.25 Write a program that can perform a letter frequency attack on an additive cipher without human intervention. Your software should produce possible plaintexts in rough order of likelihood. It would be good if your user inter- face allowed the user to specify “give me the top 10 possible plaintexts.”

3.26 Write a program that can perform a letter frequency attack on any mono- alphabetic substitution cipher without human intervention. Your software should produce possible plaintexts in rough order of likelihood. It would be good if your user interface allowed the user to specify “give me the top 10 possible plaintexts.”

3.27 Create software that can encrypt and decrypt using a 2 * 2 Hill cipher. 3.28 Create software that can perform a fast known plaintext attack on a Hill cipher,

given the dimension m. How fast are your algorithms, as a function of m?

118118

4.1 Traditional Block Cipher Structure

Stream Ciphers and Block Ciphers Motivation for the Feistel Cipher Structure The Feistel Cipher

4.2 The Data Encryption Standard

DES Encryption DES Decryption

4.3 A DES Example

Results The Avalanche Effect

4.4 The Strength of DES

The Use of 56-Bit Keys The Nature of the DES Algorithm Timing Attacks

4.5 Block Cipher Design Principles

Number of Rounds Design of Function F Key Schedule Algorithm

4.6 Key Terms, Review Questions, and Problems

CHAPTER

Block Ciphers and the Data Encryption Standard

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http://www.hiva-network.com/

4.1 / TRADITIONAL BLOCK CIPHER STRUCTURE 119

The objective of this chapter is to illustrate the principles of modern symmetric ciphers. For this purpose, we focus on the most widely used symmetric cipher: the Data Encryption Standard (DES). Although numerous symmetric ciphers have been devel- oped since the introduction of DES, and although it is destined to be replaced by the Advanced Encryption Standard (AES), DES remains the most important such algo- rithm. Furthermore, a detailed study of DES provides an understanding of the prin- ciples used in other symmetric ciphers.

This chapter begins with a discussion of the general principles of symmetric block ciphers, which are the principal type of symmetric ciphers studied in this book. The other form of symmetric ciphers, stream ciphers, are discussed in Chapter 8. Next, we cover full DES. Following this look at a specific algorithm, we return to a more general discussion of block cipher design.

Compared to public-key ciphers, such as RSA, the structure of DES and most symmetric ciphers is very complex and cannot be explained as easily as RSA and simi- lar algorithms. Accordingly, the reader may wish to begin with a simplified version of DES, which is described in Appendix G. This version allows the reader to perform encryption and decryption by hand and gain a good understanding of the working of the algorithm details. Classroom experience indicates that a study of this simplified version enhances understanding of DES.1

4.1 TRADITIONAL BLOCK CIPHER STRUCTURE

Several important symmetric block encryption algorithms in current use are based on a structure referred to as a Feistel block cipher [FEIS73]. For that reason, it is important to examine the design principles of the Feistel cipher. We begin with a comparison of stream ciphers and block ciphers. Then we discuss the motivation for the Feistel block cipher structure. Finally, we discuss some of its implications.

1However, you may safely skip Appendix G, at least on a first reading. If you get lost or bogged down in the details of DES, then you can go back and start with simplified DES.

LEARNING OBJECTIVES

After studying this chapter, you should be able to

◆ Understand the distinction between stream ciphers and block ciphers.

◆ Present an overview of the Feistel cipher and explain how decryption is the inverse of encryption.

◆ Present an overview of Data Encryption Standard (DES).

◆ Explain the concept of the avalanche effect.

◆ Discuss the cryptographic strength of DES.

◆ Summarize the principal block cipher design principles.

120 CHAPTER 4 / BLOCK CIPHERS AND THE DATA ENCRYPTION STANDARD

Stream Ciphers and Block Ciphers

A stream cipher is one that encrypts a digital data stream one bit or one byte at a time. Examples of classical stream ciphers are the autokeyed Vigenère cipher and the Vernam cipher. In the ideal case, a one-time pad version of the Vernam cipher would be used (Figure 3.7), in which the keystream (ki) is as long as the plaintext bit stream (pi). If the cryptographic keystream is random, then this cipher is unbreakable by any means other than acquiring the keystream. However, the keystream must be provided to both users in advance via some independent and secure channel. This introduces insurmountable logistical problems if the intended data traffic is very large.

Accordingly, for practical reasons, the bit-stream generator must be imple- mented as an algorithmic procedure, so that the cryptographic bit stream can be produced by both users. In this approach (Figure 4.1a), the bit-stream generator is a key-controlled algorithm and must produce a bit stream that is cryptographically strong. That is, it must be computationally impractical to predict future portions of the bit stream based on previous portions of the bit stream. The two users need only share the generating key, and each can produce the keystream.

A block cipher is one in which a block of plaintext is treated as a whole and used to produce a ciphertext block of equal length. Typically, a block size of 64 or

Figure 4.1 Stream Cipher and Block Cipher

Bit-stream generation algorithm

ENCRYPTION

(a) Stream cipher using algorithmic bit-stream generator

(b) Block cipher

Key ( K )

Encryption algorithm

Plaintext

b bits

b bits

Key ( K )

ki

Plaintext (pi)

Plaintext (pi)

Bit-stream generation algorithm

DECRYPTION

Key ( K )

ki

Ciphertext (ci)

Ciphertext

Decryption algorithm

Ciphertext

b bits

b bits

Key ( K )

Plaintext

4.1 / TRADITIONAL BLOCK CIPHER STRUCTURE 121

128 bits is used. As with a stream cipher, the two users share a symmetric encryption key (Figure 4.1b). Using some of the modes of operation explained in Chapter 7, a block cipher can be used to achieve the same effect as a stream cipher.

Far more effort has gone into analyzing block ciphers. In general, they seem applicable to a broader range of applications than stream ciphers. The vast majority of network-based symmetric cryptographic applications make use of block ciphers. Accordingly, the concern in this chapter, and in our discussions throughout the book of symmetric encryption, will primarily focus on block ciphers.

Motivation for the Feistel Cipher Structure

A block cipher operates on a plaintext block of n bits to produce a ciphertext block of n bits. There are 2n possible different plaintext blocks and, for the encryption to be reversible (i.e., for decryption to be possible), each must produce a unique ciphertext block. Such a transformation is called reversible, or nonsingular. The fol- lowing examples illustrate nonsingular and singular transformations for n = 2.

Reversible Mapping Irreversible Mapping

Plaintext Ciphertext Plaintext Ciphertext

00 11 00 11

01 10 01 10

10 00 10 01

11 01 11 01

In the latter case, a ciphertext of 01 could have been produced by one of two plain- text blocks. So if we limit ourselves to reversible mappings, the number of different transformations is 2n!.2

Figure 4.2 illustrates the logic of a general substitution cipher for n = 4. A 4-bit input produces one of 16 possible input states, which is mapped by the sub- stitution cipher into a unique one of 16 possible output states, each of which is repre- sented by 4 ciphertext bits. The encryption and decryption mappings can be defined by a tabulation, as shown in Table 4.1. This is the most general form of block cipher and can be used to define any reversible mapping between plaintext and ciphertext. Feistel refers to this as the ideal block cipher, because it allows for the maximum number of possible encryption mappings from the plaintext block [FEIS75].

But there is a practical problem with the ideal block cipher. If a small block size, such as n = 4, is used, then the system is equivalent to a classical substitution cipher. Such systems, as we have seen, are vulnerable to a statistical analysis of the plaintext. This weakness is not inherent in the use of a substitution cipher but rather results from the use of a small block size. If n is sufficiently large and an arbitrary reversible substitution between plaintext and ciphertext is allowed, then the statisti- cal characteristics of the source plaintext are masked to such an extent that this type of cryptanalysis is infeasible.

2The reasoning is as follows: For the first plaintext, we can choose any of 2n ciphertext blocks. For the second plaintext, we choose from among 2n – 1 remaining ciphertext blocks, and so on.

122 CHAPTER 4 / BLOCK CIPHERS AND THE DATA ENCRYPTION STANDARD

An arbitrary reversible substitution cipher (the ideal block cipher) for a large block size is not practical, however, from an implementation and performance point of view. For such a transformation, the mapping itself constitutes the key. Consider again Table 4.1, which defines one particular reversible mapping from

Figure 4.2 General n-bit-n-bit Block Substitution (shown with n = 4)

4-bit input

4 to 16 decoder

16 to 4 encoder

4-bit output

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Table 4.1 Encryption and Decryption Tables for Substitution Cipher of Figure 4.2

Plaintext Ciphertext

0000 1110

0001 0100

0010 1101

0011 0001

0100 0010

0101 1111

0110 1011

0111 1000

1000 0011

1001 1010

1010 0110

1011 1100

1100 0101

1101 1001

1110 0000

1111 0111

Ciphertext Plaintext

0000 1110

0001 0011

0010 0100

0011 1000

0100 0001

0101 1100

0110 1010

0111 1111

1000 0111

1001 1101

1010 1001

1011 0110

1100 1011

1101 0010

1110 0000

1111 0101

4.1 / TRADITIONAL BLOCK CIPHER STRUCTURE 123

plaintext to ciphertext for n = 4. The mapping can be defined by the entries in the second column, which show the value of the ciphertext for each plaintext block. This, in essence, is the key that determines the specific mapping from among all possible mappings. In this case, using this straightforward method of defining the key, the required key length is (4 bits) * (16 rows) = 64 bits. In general, for an n-bit ideal block cipher, the length of the key defined in this fashion is n * 2n bits. For a 64-bit block, which is a desirable length to thwart statistical attacks, the required key length is 64 * 264 = 270 ≈ 1021 bits.

In considering these difficulties, Feistel points out that what is needed is an approximation to the ideal block cipher system for large n, built up out of compo- nents that are easily realizable [FEIS75]. But before turning to Feistel’s approach, let us make one other observation. We could use the general block substitution cipher but, to make its implementation tractable, confine ourselves to a subset of the 2n! possible reversible mappings. For example, suppose we define the mapping in terms of a set of linear equations. In the case of n = 4, we have

y1 = k11x1 + k12x2 + k13x3 + k14x4 y2 = k21x1 + k22x2 + k23x3 + k24x4 y3 = k31x1 + k32x2 + k33x3 + k34x4 y4 = k41x1 + k42x2 + k43x3 + k44x4

where the xi are the four binary digits of the plaintext block, the yi are the four bi- nary digits of the ciphertext block, the kij are the binary coefficients, and arithmetic is mod 2. The key size is just n2, in this case 16 bits. The danger with this kind of for- mulation is that it may be vulnerable to cryptanalysis by an attacker that is aware of the structure of the algorithm. In this example, what we have is essentially the Hill cipher discussed in Chapter 3, applied to binary data rather than characters. As we saw in Chapter 3, a simple linear system such as this is quite vulnerable.

The Feistel Cipher

Feistel proposed [FEIS73] that we can approximate the ideal block cipher by utiliz- ing the concept of a product cipher, which is the execution of two or more simple ciphers in sequence in such a way that the final result or product is cryptographi- cally stronger than any of the component ciphers. The essence of the approach is to develop a block cipher with a key length of k bits and a block length of n bits, allowing a total of 2k possible transformations, rather than the 2n! transformations available with the ideal block cipher.

In particular, Feistel proposed the use of a cipher that alternates substitutions and permutations, where these terms are defined as follows:

■ Substitution: Each plaintext element or group of elements is uniquely replaced by a corresponding ciphertext element or group of elements.

■ Permutation: A sequence of plaintext elements is replaced by a permutation of that sequence. That is, no elements are added or deleted or replaced in the sequence, rather the order in which the elements appear in the sequence is changed.

124 CHAPTER 4 / BLOCK CIPHERS AND THE DATA ENCRYPTION STANDARD

In fact, Feistel’s is a practical application of a proposal by Claude Shannon to develop a product cipher that alternates confusion and diffusion functions [SHAN49].3 We look next at these concepts of diffusion and confusion and then present the Feistel cipher. But first, it is worth commenting on this remarkable fact: The Feistel cipher structure, which dates back over a quarter century and which, in turn, is based on Shannon’s proposal of 1945, is the structure used by a number of significant symmetric block ciphers currently in use. In particular, the Feistel struc- ture is used for Triple Data Encryption Algorithm (TDEA), which is one of the two encryption algorithms (along with AES), approved for general use by the National Institute of Standards and Technology (NIST). The Feistel structure is also used for several schemes for format-preserving encryption, which have recently come into prominence. In addition, the Camellia block cipher is a Feistel structure; it is one of the possible symmetric ciphers in TLS and a number of other Internet security protocols. Both TDEA and format-preserving encryption are covered in Chapter 7.

DIFFUSION AND CONFUSION The terms diffusion and confusion were introduced by Claude Shannon to capture the two basic building blocks for any cryptographic sys- tem [SHAN49]. Shannon’s concern was to thwart cryptanalysis based on statisti- cal analysis. The reasoning is as follows. Assume the attacker has some knowledge of the statistical characteristics of the plaintext. For example, in a human-readable message in some language, the frequency distribution of the various letters may be known. Or there may be words or phrases likely to appear in the message (probable words). If these statistics are in any way reflected in the ciphertext, the cryptanalyst may be able to deduce the encryption key, part of the key, or at least a set of keys likely to contain the exact key. In what Shannon refers to as a strongly ideal cipher, all statistics of the ciphertext are independent of the particular key used. The arbi- trary substitution cipher that we discussed previously (Figure 4.2) is such a cipher, but as we have seen, it is impractical.4

Other than recourse to ideal systems, Shannon suggests two methods for frustrating statistical cryptanalysis: diffusion and confusion. In diffusion, the sta- tistical structure of the plaintext is dissipated into long-range statistics of the ciphertext. This is achieved by having each plaintext digit affect the value of many ciphertext digits; generally, this is equivalent to having each ciphertext digit be affected by many plaintext digits. An example of diffusion is to encrypt a message M = m1, m2, m3, c of characters with an averaging operation:

yn = ¢ ak i=1

mn + i≤ mod 26 3The paper is available at box.com/Crypto7e. Shannon’s 1949 paper appeared originally as a classified report in 1945. Shannon enjoys an amazing and unique position in the history of computer and informa- tion science. He not only developed the seminal ideas of modern cryptography but is also responsible for inventing the discipline of information theory. Based on his work in information theory, he developed a formula for the capacity of a data communications channel, which is still used today. In addition, he founded another discipline, the application of Boolean algebra to the study of digital circuits; this last he managed to toss off as a master’s thesis. 4Appendix F expands on Shannon’s concepts concerning measures of secrecy and the security of crypto- graphic algorithms.

4.1 / TRADITIONAL BLOCK CIPHER STRUCTURE 125

adding k successive letters to get a ciphertext letter yn. One can show that the sta- tistical structure of the plaintext has been dissipated. Thus, the letter frequencies in the ciphertext will be more nearly equal than in the plaintext; the digram frequen- cies will also be more nearly equal, and so on. In a binary block cipher, diffusion can be achieved by repeatedly performing some permutation on the data followed by applying a function to that permutation; the effect is that bits from different posi- tions in the original plaintext contribute to a single bit of ciphertext.5

Every block cipher involves a transformation of a block of plaintext into a block of ciphertext, where the transformation depends on the key. The mechanism of diffusion seeks to make the statistical relationship between the plaintext and ciphertext as complex as possible in order to thwart attempts to deduce the key. On the other hand, confusion seeks to make the relationship between the statistics of the ciphertext and the value of the encryption key as complex as possible, again to thwart attempts to discover the key. Thus, even if the attacker can get some handle on the statistics of the ciphertext, the way in which the key was used to produce that ciphertext is so complex as to make it difficult to deduce the key. This is achieved by the use of a complex substitution algorithm. In contrast, a simple linear substitution function would add little confusion.

As [ROBS95b] points out, so successful are diffusion and confusion in captur- ing the essence of the desired attributes of a block cipher that they have become the cornerstone of modern block cipher design.

FEISTEL CIPHER STRUCTURE The left-hand side of Figure 4.3 depicts the encryption structure proposed by Feistel. The inputs to the encryption algorithm are a plaintext block of length 2w bits and a key K. The plaintext block is divided into two halves, LE0 and RE0. The two halves of the data pass through n rounds of processing and then combine to produce the ciphertext block. Each round i has as inputs LEi- 1 and REi- 1 derived from the previous round, as well as a subkey Ki derived from the over- all K. In general, the subkeys Ki are different from K and from each other. In Figure 4.3, 16 rounds are used, although any number of rounds could be implemented.

All rounds have the same structure. A substitution is performed on the left half of the data. This is done by applying a round function F to the right half of the data and then taking the exclusive-OR of the output of that function and the left half of the data. The round function has the same general structure for each round but is parameterized by the round subkey Ki. Another way to express this is to say that F is a function of right-half block of w bits and a subkey of y bits, which pro- duces an output value of length w bits: F(REi, Ki+ 1). Following this substitution, a permutation is performed that consists of the interchange of the two halves of the data.6 This structure is a particular form of the substitution-permutation network (SPN) proposed by Shannon.

5Some books on cryptography equate permutation with diffusion. This is incorrect. Permutation, by itself, does not change the statistics of the plaintext at the level of individual letters or permuted blocks. For exam- ple, in DES, the permutation swaps two 32-bit blocks, so statistics of strings of 32 bits or less are preserved. 6The final round is followed by an interchange that undoes the interchange that is part of the final round. One could simply leave both interchanges out of the diagram, at the sacrifice of some consistency of pre- sentation. In any case, the effective lack of a swap in the final round is done to simplify the implementa- tion of the decryption process, as we shall see.

126 CHAPTER 4 / BLOCK CIPHERS AND THE DATA ENCRYPTION STANDARD

The exact realization of a Feistel network depends on the choice of the follow- ing parameters and design features:

■ Block size: Larger block sizes mean greater security (all other things being equal) but reduced encryption/decryption speed for a given algorithm. The greater security is achieved by greater diffusion. Traditionally, a block size of 64 bits has been considered a reasonable tradeoff and was nearly universal in block cipher design. However, the new AES uses a 128-bit block size.

Figure 4.3 Feistel Encryption and Decryption (16 rounds)

Output (ciphertext)

K1

LD0 = RE16 RD0 = LE16

LD2 = RE14 RD2 = LE14

LD14 = RE2 RD14 = LE2

LD16 = RE0

LD17 = RE0

RD16 = LE0

RD17 = LE0

RD1 = LE15LD1 = RE15

RD15 = LE1LD15 = RE1

Input (ciphertext)

Output (plaintext)

R ou

nd 1

K1

K2

K15

K16

K2

K15

K16

F

LE0 RE0

Input (plaintext)

LE1 RE1

LE2 RE2

F

F

LE14 RE14

LE15 RE15

LE16 RE16

LE17 RE17

F

F

F

F

F

R ou

nd 2

R ou

nd 1

5 R

ou nd

1 6

R ou

nd 1

6 R

ou nd

1 5

R ou

nd 2

R ou

nd 1

4.1 / TRADITIONAL BLOCK CIPHER STRUCTURE 127

■ Key size: Larger key size means greater security but may decrease encryption/ decryption speed. The greater security is achieved by greater resistance to brute-force attacks and greater confusion. Key sizes of 64 bits or less are now widely considered to be inadequate, and 128 bits has become a common size.

■ Number of rounds: The essence of the Feistel cipher is that a single round offers inadequate security but that multiple rounds offer increasing security. A typical size is 16 rounds.

■ Subkey generation algorithm: Greater complexity in this algorithm should lead to greater difficulty of cryptanalysis.

■ Round function F: Again, greater complexity generally means greater resis- tance to cryptanalysis.

There are two other considerations in the design of a Feistel cipher:

■ Fast software encryption/decryption: In many cases, encryption is embedded in applications or utility functions in such a way as to preclude a hardware im- plementation. Accordingly, the speed of execution of the algorithm becomes a concern.

■ Ease of analysis: Although we would like to make our algorithm as difficult as possible to cryptanalyze, there is great benefit in making the algorithm easy to analyze. That is, if the algorithm can be concisely and clearly explained, it is easier to analyze that algorithm for cryptanalytic vulnerabilities and therefore develop a higher level of assurance as to its strength. DES, for example, does not have an easily analyzed functionality.

FEISTEL DECRYPTION ALGORITHM The process of decryption with a Feistel cipher is essentially the same as the encryption process. The rule is as follows: Use the ciphertext as input to the algorithm, but use the subkeys Ki in reverse order. That is, use Kn in the first round, Kn – 1 in the second round, and so on, until K1 is used in the last round. This is a nice feature, because it means we need not implement two different algorithms; one for encryption and one for decryption.

To see that the same algorithm with a reversed key order produces the cor- rect result, Figure 4.3 shows the encryption process going down the left-hand side and the decryption process going up the right-hand side for a 16-round algorithm. For clarity, we use the notation LEi and REi for data traveling through the encryp- tion algorithm and LDi and RDi for data traveling through the decryption algo- rithm. The diagram indicates that, at every round, the intermediate value of the decryption process is equal to the corresponding value of the encryption process with the two halves of the value swapped. To put this another way, let the output of the ith encryption round be LEi ‘REi (LEi concatenated with REi). Then the cor- responding output of the (16 – i)th decryption round is REi ‘LEi or, equivalently, LD16 – i ‘RD16 – i.

Let us walk through Figure 4.3 to demonstrate the validity of the preceding assertions. After the last iteration of the encryption process, the two halves of the output are swapped, so that the ciphertext is RE16 ‘LE16. The output of that round is the ciphertext. Now take that ciphertext and use it as input to the same algorithm. The input to the first round is RE16 ‘LE16, which is equal to the 32-bit swap of the output of the sixteenth round of the encryption process.

Hiva-Network.Com

http://www.hiva-network.com/

128 CHAPTER 4 / BLOCK CIPHERS AND THE DATA ENCRYPTION STANDARD

Now we would like to show that the output of the first round of the decryption process is equal to a 32-bit swap of the input to the sixteenth round of the encryp- tion process. First, consider the encryption process. We see that

LE16 = RE15 RE16 = LE15⊕ F(RE15, K16)

On the decryption side,

LD1 = RD0 = LE16 = RE15 RD1 = LD0⊕ F(RD0, K16)

= RE16⊕ F(RE15, K16) = [LE15⊕ F(RE15, K16)]⊕ F(RE15, K16)

The XOR has the following properties:

[A⊕ B]⊕ C = A⊕ [B⊕ C] D⊕D = 0 E⊕ 0 = E

Thus, we have LD1 = RE15 and RD1 = LE15. Therefore, the output of the first round of the decryption process is RE15 ‘LE15, which is the 32-bit swap of the input to the sixteenth round of the encryption. This correspondence holds all the way through the 16 iterations, as is easily shown. We can cast this process in general terms. For the ith iteration of the encryption algorithm,

LEi = REi- 1 REi = LEi- 1⊕ F(REi- 1, Ki)

Rearranging terms:

REi- 1 = LEi LEi- 1 = REi⊕ F(REi- 1, Ki) = REi⊕ F(LEi, Ki)

Thus, we have described the inputs to the ith iteration as a function of the outputs, and these equations confirm the assignments shown in the right-hand side of Figure 4.3.

Finally, we see that the output of the last round of the decryption process is RE0 ‘LE0. A 32-bit swap recovers the original plaintext, demonstrating the validity of the Feistel decryption process.

Note that the derivation does not require that F be a reversible function. To see this, take a limiting case in which F produces a constant output (e.g., all ones) regardless of the values of its two arguments. The equations still hold.

To help clarify the preceding concepts, let us look at a specific example (Figure 4.4 and focus on the fifteenth round of encryption, corresponding to the sec- ond round of decryption. Suppose that the blocks at each stage are 32 bits (two 16-bit halves) and that the key size is 24 bits. Suppose that at the end of encryption round fourteen, the value of the intermediate block (in hexadecimal) is DE7F03A6. Then LE14 = DE7F and RE14 = 03A6. Also assume that the value of K15 is 12DE52. After round 15, we have LE15 = 03A6 and RE15 = F(03A6, 12DE52)⊕DE7F.

4.2 / THE DATA ENCRYPTION STANDARD 129

Now let’s look at the decryption. We assume that LD1 = RE15 and RD1 = LE15, as shown in Figure 4.3, and we want to demonstrate that LD2 = RE14 and RD2 = LE14. So, we start with LD1 = F(03A6, 12DE52)⊕DE7F and RD1 = 03A6. Then, from Figure 4.3, LD2 = 03A6 = RE14 and RD2 = F(03A6, 12DE52)⊕ [F(03A6, 12DE52)⊕DE7F] = DE7F = LE14.

4.2 THE DATA ENCRYPTION STANDARD

Until the introduction of the Advanced Encryption Standard (AES) in 2001, the Data Encryption Standard (DES) was the most widely used encryption scheme. DES was issued in 1977 by the National Bureau of Standards, now the National Institute of Standards and Technology (NIST), as Federal Information Processing Standard 46 (FIPS PUB 46). The algorithm itself is referred to as the Data Encryption Algorithm (DEA).7 For DEA, data are encrypted in 64-bit blocks using a 56-bit key. The algorithm transforms 64-bit input in a series of steps into a 64-bit output. The same steps, with the same key, are used to reverse the encryption.

Over the years, DES became the dominant symmetric encryption algorithm, especially in financial applications. In 1994, NIST reaffirmed DES for federal use for another five years; NIST recommended the use of DES for applications other than the protection of classified information. In 1999, NIST issued a new version of its standard (FIPS PUB 46-3) that indicated that DES should be used only for legacy systems and that triple DES (which in essence involves repeating the DES algorithm three times on the plaintext using two or three different keys to produce the ciphertext) be used. We study triple DES in Chapter 7. Because the underlying encryption and decryption algorithms are the same for DES and triple DES, it remains important to understand the DES cipher. This section provides an overview.For the interested reader, Appendix S provides further detail.

7The terminology is a bit confusing. Until recently, the terms DES and DEA could be used interchange- ably. However, the most recent edition of the DES document includes a specification of the DEA described here plus the triple DEA (TDEA) described in Chapter 7. Both DEA and TDEA are part of the Data Encryption Standard. Further, until the recent adoption of the official term TDEA, the triple DEA algorithm was typically referred to as triple DES and written as 3DES. For the sake of convenience, we will use the term 3DES.

Figure 4.4 Feistel Example

12DE52

12DE52

F

DE7F 03A6

Decryption roundEncryption round

03A6

6A306A30 F(03A6, 12DE52) DE7F F(03A6, 12DE52) DE7F

F(03A6, 12DE52) [F(03A6, 12DE52) DE7F]

= DE7F

FR ou

nd 1

5

R ou

nd 2

130 CHAPTER 4 / BLOCK CIPHERS AND THE DATA ENCRYPTION STANDARD

DES Encryption

The overall scheme for DES encryption is illustrated in Figure 4.5. As with any encryption scheme, there are two inputs to the encryption function: the plaintext to be encrypted and the key. In this case, the plaintext must be 64 bits in length and the key is 56 bits in length.8

Looking at the left-hand side of the figure, we can see that the processing of the plaintext proceeds in three phases. First, the 64-bit plaintext passes through an initial permutation (IP) that rearranges the bits to produce the permuted input.

8Actually, the function expects a 64-bit key as input. However, only 56 of these bits are ever used; the other 8 bits can be used as parity bits or simply set arbitrarily.

Figure 4.5 General Depiction of DES Encryption Algorithm

Initial permutation

Permuted choice 2Round 1

32-bit swap

Inverse initial permutation

Permuted choice 1

Round 2

Round 16

64-bit plaintext 64-bit key

K1

K2

K16

64-bit ciphertext

Left circular shift

Permuted choice 2 Left circular shift

Permuted choice 2 Left circular shift

64 56

56

56

56

48

48

48

56 64

64 bits

4.3 / A DES EXAMPLE 131

This is followed by a phase consisting of sixteen rounds of the same function, which involves both permutation and substitution functions. The output of the last (six- teenth) round consists of 64 bits that are a function of the input plaintext and the key. The left and right halves of the output are swapped to produce the preoutput. Finally, the preoutput is passed through a permutation [IP-1] that is the inverse of the initial permutation function, to produce the 64-bit ciphertext. With the excep- tion of the initial and final permutations, DES has the exact structure of a Feistel cipher, as shown in Figure 4.3.

The right-hand portion of Figure 4.5 shows the way in which the 56-bit key is used. Initially, the key is passed through a permutation function. Then, for each of the sixteen rounds, a subkey (Ki) is produced by the combination of a left circular shift and a permutation. The permutation function is the same for each round, but a different subkey is produced because of the repeated shifts of the key bits.

DES Decryption

As with any Feistel cipher, decryption uses the same algorithm as encryption, except that the application of the subkeys is reversed. Additionally, the initial and final permutations are reversed.

4.3 A DES EXAMPLE

We now work through an example and consider some of its implications. Although you are not expected to duplicate the example by hand, you will find it informative to study the hex patterns that occur from one step to the next.

For this example, the plaintext is a hexadecimal palindrome. The plaintext, key, and resulting ciphertext are as follows:

Plaintext: 02468aceeca86420

Key: 0f1571c947d9e859

Ciphertext: da02ce3a89ecac3b

Results

Table 4.2 shows the progression of the algorithm. The first row shows the 32-bit values of the left and right halves of data after the initial permutation. The next 16 rows show the results after each round. Also shown is the value of the 48-bit subkey generated for each round. Note that Li = Ri- 1. The final row shows the left- and right-hand values after the inverse initial permutation. These two values combined form the ciphertext.

The Avalanche Effect

A desirable property of any encryption algorithm is that a small change in either the plaintext or the key should produce a significant change in the ciphertext. In particular, a change in one bit of the plaintext or one bit of the key should produce

132 CHAPTER 4 / BLOCK CIPHERS AND THE DATA ENCRYPTION STANDARD

a change in many bits of the ciphertext. This is referred to as the avalanche effect. If the change were small, this might provide a way to reduce the size of the plaintext or key space to be searched.

Using the example from Table 4.2, Table 4.3 shows the result when the fourth bit of the plaintext is changed, so that the plaintext is 12468aceeca86420. The second column of the table shows the intermediate 64-bit values at the end of each round for the two plaintexts. The third column shows the number of bits that differ between the two intermediate values. The table shows that, after just three rounds, 18 bits differ between the two blocks. On completion, the two ciphertexts differ in 32 bit positions.

Table 4.4 shows a similar test using the original plaintext of with two keys that differ in only the fourth bit position: the original key, 0f1571c947d9e859, and the altered key, 1f1571c947d9e859. Again, the results show that about half of the bits in the ciphertext differ and that the avalanche effect is pronounced after just a few rounds.

Round Ki Li Ri

IP 5a005a00 3cf03c0f

1 1e030f03080d2930 3cf03c0f bad22845

2 0a31293432242318 bad22845 99e9b723

3 23072318201d0c1d 99e9b723 0bae3b9e

4 05261d3824311a20 0bae3b9e 42415649

5 3325340136002c25 42415649 18b3fa41

6 123a2d0d04262a1c 18b3fa41 9616fe23

7 021f120b1c130611 9616fe23 67117cf2

8 1c10372a2832002b 67117cf2 c11bfc09

9 04292a380c341f03 c11bfc09 887fbc6c

10 2703212607280403 887fbc6c 600f7e8b

11 2826390c31261504 600f7e8b f596506e

12 12071c241a0a0f08 f596506e 738538b8

13 300935393c0d100b 738538b8 c6a62c4e

14 311e09231321182a c6a62c4e 56b0bd75

15 283d3e0227072528 56b0bd75 75e8fd8f

16 2921080b13143025 75e8fd8f 25896490

IP−1 da02ce3a 89ecac3b

Note: DES subkeys are shown as eight 6-bit values in hex format

Table 4.2 DES Example

4.3 / A DES EXAMPLE 133

Table 4.3 Avalanche Effect in DES: Change in Plaintext

Round D

9 c11bfc09887fbc6c 99f911532eed7d94

32

10 887fbc6c600f7e8b 2eed7d94d0f23094

34

11 600f7e8bf596506e d0f23094455da9c4

37

12 f596506e738538b8 455da9c47f6e3cf3

31

13 738538b8c6a62c4e 7f6e3cf34bc1a8d9

29

14 c6a62c4e56b0bd75 4bc1a8d91e07d409

33

15 56b0bd7575e8fd8f 1e07d4091ce2e6dc

31

16 75e8fd8f25896490 1ce2e6dc365e5f59

32

IP−1 da02ce3a89ecac3b 057cde97d7683f2a

32

Round D

02468aceeca86420 12468aceeca86420

1

1 3cf03c0fbad22845 3cf03c0fbad32845

1

2 bad2284599e9b723 bad3284539a9b7a3

5

3 99e9b7230bae3b9e 39a9b7a3171cb8b3

18

4 0bae3b9e42415649 171cb8b3ccaca55e

34

5 4241564918b3fa41 ccaca55ed16c3653

37

6 18b3fa419616fe23 d16c3653cf402c68

33

7 9616fe2367117cf2 cf402c682b2cefbc

32

8 67117cf2c11bfc09 2b2cefbc99f91153

33

Table 4.4 Avalanche Effect in DES: Change in Key

Round D

02468aceeca86420 02468aceeca86420

0

1 3cf03c0fbad22845 3cf03c0f9ad628c5

3

2 bad2284599e9b723 9ad628c59939136b

11

3 99e9b7230bae3b9e 9939136b768067b7

25

4 0bae3b9e42415649 768067b75a8807c5

29

5 4241564918b3fa41 5a8807c5488dbe94

26

6 18b3fa419616fe23 488dbe94aba7fe53

26

7 9616fe2367117cf2 aba7fe53177d21e4

27

8 67117cf2c11bfc09 177d21e4548f1de4

32

Round D

9 c11bfc09887fbc6c 548f1de471f64dfd

34

10 887fbc6c600f7e8b 71f64dfd4279876c

36

11 600f7e8bf596506e 4279876c399fdc0d

32

12 f596506e738538b8 399fdc0d6d208dbb

28

13 738538b8c6a62c4e 6d208dbbb9bdeeaa

33

14 c6a62c4e56b0bd75 b9bdeeaad2c3a56f

30

15 56b0bd7575e8fd8f d2c3a56f2765c1fb

27

16 75e8fd8f25896490 2765c1fb01263dc4

30

IP−1 da02ce3a89ecac3b ee92b50606b62b0b

30

134 CHAPTER 4 / BLOCK CIPHERS AND THE DATA ENCRYPTION STANDARD

4.4 THE STRENGTH OF DES

Since its adoption as a federal standard, there have been lingering concerns about the level of security provided by DES. These concerns, by and large, fall into two areas: key size and the nature of the algorithm.

The Use of 56-Bit Keys

With a key length of 56 bits, there are 256 possible keys, which is approximately 7.2 * 1016 keys. Thus, on the face of it, a brute-force attack appears impractical. Assuming that, on average, half the key space has to be searched, a single machine performing one DES encryption per microsecond would take more than a thousand years to break the cipher.

However, the assumption of one encryption per microsecond is overly con- servative. As far back as 1977, Diffie and Hellman postulated that the technology existed to build a parallel machine with 1 million encryption devices, each of which could perform one encryption per microsecond [DIFF77]. This would bring the average search time down to about 10 hours. The authors estimated that the cost would be about $20 million in 1977 dollars.

With current technology, it is not even necessary to use special, purpose-built hardware. Rather, the speed of commercial, off-the-shelf processors threaten the security of DES. A recent paper from Seagate Technology [SEAG08] suggests that a rate of 1 billion (109) key combinations per second is reasonable for today’s mul- ticore computers. Recent offerings confirm this. Both Intel and AMD now offer hardware-based instructions to accelerate the use of AES. Tests run on a contem- porary multicore Intel machine resulted in an encryption rate of about half a bil- lion encryptions per second [BASU12]. Another recent analysis suggests that with contemporary supercomputer technology, a rate of 1013 encryptions per second is reasonable [AROR12].

With these results in mind, Table 4.5 shows how much time is required for a brute-force attack for various key sizes. As can be seen, a single PC can break DES in about a year; if multiple PCs work in parallel, the time is drastically shortened. And today’s supercomputers should be able to find a key in about an hour. Key sizes of 128 bits or greater are effectively unbreakable using simply a brute-force approach. Even if we managed to speed up the attacking system by a factor of 1 trillion (1012), it would still take over 100,000 years to break a code using a 128-bit key.

Fortunately, there are a number of alternatives to DES, the most important of which are AES and triple DES, discussed in Chapters 6 and 7, respectively.

The Nature of the DES Algorithm

Another concern is the possibility that cryptanalysis is possible by exploiting the characteristics of the DES algorithm. The focus of concern has been on the eight substitution tables, or S-boxes, that are used in each iteration (described in Appendix S). Because the design criteria for these boxes, and indeed for the entire algorithm, were not made public, there is a suspicion that the boxes were con- structed in such a way that cryptanalysis is possible for an opponent who knows

4.5 / BLOCK CIPHER DESIGN PRINCIPLES 135

Key Size (bits) Cipher

Number of Alternative

Keys Time Required at 109

Decryptions/s

Time Required at 1013

Decryptions/s

56 DES 256 ≈ 7.2 * 1016 255 ns = 1.125 years 1 hour

128 AES 2128 ≈ 3.4 * 1038 2127 ns = 5.3 * 1021 years 5.3 * 1017 years

168 Triple DES 2168 ≈ 3.7 * 1050 2167 ns = 5.8 * 1033 years 5.8 * 1029 years

192 AES 2192 ≈ 6.3 * 1057 2191 ns = 9.8 * 1040 years 9.8 * 1036 years

256 AES 2256 ≈ 1.2 * 1077 2255 ns = 1.8 * 1060 years 1.8 * 1056 years

26 characters (permutation)

Monoalphabetic 2! = 4 * 1026 2 * 1026 ns = 6.3 * 109 years 6.3 * 106 years

Table 4.5 Average Time Required for Exhaustive Key Search

the weaknesses in the S-boxes. This assertion is tantalizing, and over the years a number of regularities and unexpected behaviors of the S-boxes have been discov- ered. Despite this, no one has so far succeeded in discovering the supposed fatal weaknesses in the S-boxes.9

Timing Attacks

We discuss timing attacks in more detail in Part Two, as they relate to public-key algorithms. However, the issue may also be relevant for symmetric ciphers. In essence, a timing attack is one in which information about the key or the plaintext is obtained by observing how long it takes a given implementation to perform decryp- tions on various ciphertexts. A timing attack exploits the fact that an encryption or decryption algorithm often takes slightly different amounts of time on different inputs. [HEVI99] reports on an approach that yields the Hamming weight (number of bits equal to one) of the secret key. This is a long way from knowing the actual key, but it is an intriguing first step. The authors conclude that DES appears to be fairly resistant to a successful timing attack but suggest some avenues to explore. Although this is an interesting line of attack, it so far appears unlikely that this tech- nique will ever be successful against DES or more powerful symmetric ciphers such as triple DES and AES.

4.5 BLOCK CIPHER DESIGN PRINCIPLES

Although much progress has been made in designing block ciphers that are cryp- tographically strong, the basic principles have not changed all that much since the work of Feistel and the DES design team in the early 1970s. In this section we look at three critical aspects of block cipher design: the number of rounds, design of the function F, and key scheduling.

9At least, no one has publicly acknowledged such a discovery.

136 CHAPTER 4 / BLOCK CIPHERS AND THE DATA ENCRYPTION STANDARD

Number of Rounds

The cryptographic strength of a Feistel cipher derives from three aspects of the design: the number of rounds, the function F, and the key schedule algorithm. Let us look first at the choice of the number of rounds.

The greater the number of rounds, the more difficult it is to perform crypt- analysis, even for a relatively weak F. In general, the criterion should be that the number of rounds is chosen so that known cryptanalytic efforts require greater effort than a simple brute-force key search attack. This criterion was certainly used in the design of DES. Schneier [SCHN96] observes that for 16-round DES, a dif- ferential cryptanalysis attack is slightly less efficient than brute force: The differen- tial cryptanalysis attack requires 255.1 operations,10 whereas brute force requires 255. If DES had 15 or fewer rounds, differential cryptanalysis would require less effort than a brute-force key search.

This criterion is attractive, because it makes it easy to judge the strength of an algorithm and to compare different algorithms. In the absence of a cryptana- lytic breakthrough, the strength of any algorithm that satisfies the criterion can be judged solely on key length.

Design of Function F

The heart of a Feistel block cipher is the function F, which provides the element of confusion in a Feistel cipher. Thus, it must be difficult to “unscramble” the substitu- tion performed by F. One obvious criterion is that F be nonlinear, as we discussed previously. The more nonlinear F, the more difficult any type of cryptanalysis will be. There are several measures of nonlinearity, which are beyond the scope of this book. In rough terms, the more difficult it is to approximate F by a set of linear equations, the more nonlinear F is.

Several other criteria should be considered in designing F. We would like the algorithm to have good avalanche properties. Recall that, in general, this means that a change in one bit of the input should produce a change in many bits of the output. A more stringent version of this is the strict avalanche criterion (SAC) [WEBS86], which states that any output bit j of an S-box (see Appendix S for a discussion of S-boxes) should change with probability 1/2 when any single input bit i is inverted for all i, j. Although SAC is expressed in terms of S-boxes, a similar criterion could be applied to F as a whole. This is important when considering designs that do not include S-boxes.

Another criterion proposed in [WEBS86] is the bit independence criterion (BIC), which states that output bits j and k should change independently when any single input bit i is inverted for all i, j, and k. The SAC and BIC criteria appear to strengthen the effectiveness of the confusion function.

10Differential cryptanalysis of DES requires 247 chosen plaintext. If all you have to work with is known plaintext, then you must sort through a large quantity of known plaintext–ciphertext pairs looking for the useful ones. This brings the level of effort up to 255.1.

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4.6 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 137

Key Schedule Algorithm

With any Feistel block cipher, the key is used to generate one subkey for each round. In general, we would like to select subkeys to maximize the difficulty of deducing individual subkeys and the difficulty of working back to the main key. No general principles for this have yet been promulgated.

Adams suggests [ADAM94] that, at minimum, the key schedule should guar- antee key/ciphertext Strict Avalanche Criterion and Bit Independence Criterion.

4.6 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS

Key Terms

avalanche effect block cipher confusion Data Encryption Standard

(DES) diffusion

Feistel cipher irreversible mapping key permutation product cipher reversible mapping

round round function subkey substitution

Review Questions 4.1 Briefly define a nonsingular transformation. 4.2 What is the difference between a block cipher and a stream cipher? 4.3 Why is it not practical to use an arbitrary reversible substitution cipher of the kind

shown in Table 4.1? 4.4 Briefly define the terms substitution and permutation. 4.5 What is the difference between diffusion and confusion? 4.6 Which parameters and design choices determine the actual algorithm of a Feistel

cipher? 4.7 What are the critical aspects of Feistel cipher design?

Problems

4.1 a. In Section 4.1, under the subsection on the motivation for the Feistel cipher struc- ture, it was stated that, for a block of n bits, the number of different reversible mappings for the ideal block cipher is 2n!. Justify.

b. In that same discussion, it was stated that for the ideal block cipher, which allows all possible reversible mappings, the size of the key is n * 2n bits. But, if there are 2n! possible mappings, it should take log2 2

n! bits to discriminate among the different mappings, and so the key length should be log2 2

n!. However, log2 2 n! 6 n * 2n.

Explain the discrepancy.

138 CHAPTER 4 / BLOCK CIPHERS AND THE DATA ENCRYPTION STANDARD

4.2 Consider a Feistel cipher composed of sixteen rounds with a block length of 128 bits and a key length of 128 bits. Suppose that, for a given k, the key scheduling algorithm determines values for the first eight round keys, k1, k2, c k8, and then sets

k9 = k8, k10 = k7, k11 = k6, c , k16 = k1

Suppose you have a ciphertext c. Explain how, with access to an encryption oracle, you can decrypt c and determine m using just a single oracle query. This shows that such a cipher is vulnerable to a chosen plaintext attack. (An encryption oracle can be thought of as a device that, when given a plaintext, returns the corresponding cipher- text. The internal details of the device are not known to you and you cannot break open the device. You can only gain information from the oracle by making queries to it and observing its responses.)

4.3 Let p be a permutation of the integers 0, 1, 2, c , (2n – 1), such that p(m) gives the permuted value of m, 0 … m 6 2n. Put another way, p maps the set of n-bit integers into itself and no two integers map into the same integer. DES is such a permutation for 64-bit integers. We say that p has a fixed point at m if p(m) = m. That is, if p is an encryption mapping, then a fixed point corresponds to a message that encrypts to itself. We are interested in the number of fixed points in a randomly chosen permuta- tion p. Show the somewhat unexpected result that the number of fixed points for p is 1 on an average, and this number is independent of the size of the permutation.

4.4 Consider a block encryption algorithm that encrypts blocks of length n, and let N = 2n. Say we have t plaintext–ciphertext pairs Pi, Ci = E(K, Pi), where we assume that the key K selects one of the N! possible mappings. Imagine that we wish to find K by exhaustive search. We could generate key K′ and test whether Ci = E(K′, Pi) for 1 … i … t. If K′ encrypts each Pi to its proper Ci, then we have evidence that K = K′. However, it may be the case that the mappings E(K, # ) and E(K′, # ) exactly agree on the t plaintext–cipher text pairs Pi, Ci and agree on no other pairs. a. What is the probability that E(K, # ) and E(K′, # ) are in fact distinct mappings? b. What is the probability that E(K, # ) and E(K′, # ) agree on another t′ plaintext–

ciphertext pairs where 0 … t′ … N – t? 4.5 For any block cipher, the fact that it is a nonlinear function is crucial to its security. To

see this, suppose that we have a linear block cipher EL that encrypts 256-bit blocks of plaintext into 256-bit blocks of ciphertext. Let EL(k, m) denote the encryption of a 256-bit message m under a key k (the actual bit length of k is irrelevant). Thus,

EL(k, [m1⊕ m2]) = EL(k, m1)⊕ EL(k, m2) for all 128@bit patterns m1, m2.

Describe how, with 256 chosen ciphertexts, an adversary can decrypt any ciphertext without knowledge of the secret key k. (A “chosen ciphertext” means that an adver- sary has the ability to choose a ciphertext and then obtain its decryption. Here, you have 256 plaintext/ciphertext pairs to work with and you have the ability to choose the value of the ciphertexts.)

4.6 Suppose the DES F function mapped every 32-bit input R, regardless of the value of the input K, to; a. 32-bit string of zero b. R

Then 1. What function would DES then compute? 2. What would the decryption look like?

Hint: Use the following properties of the XOR operation:

(A⊕ B)⊕ C = A⊕ (B⊕ C) (A⊕ A) = 0 (A⊕ 0 ) = A

A⊕ 1 = bitwise complement of A

4.6 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 139

where A,B,C are n-bit strings of bits 0 is an n-bit string of zeros 1 is an n-bit string of one

4.7 Show that DES decryption is, in fact, the inverse of DES encryption. 4.8 The 32-bit swap after the sixteenth iteration of the DES algorithm is needed to make

the encryption process invertible by simply running the ciphertext back through the algorithm with the key order reversed. This was demonstrated in the preceding prob- lem. However, it still may not be entirely clear why the 32-bit swap is needed. To demonstrate why, solve the following exercises. First, some notation:

A ‘B = the concatenation of the bit strings A and B Ti(R ‘L) = the transformation defined by the ith iteration of the encryption

algorithm for 1 … I … 16 TDi(R ‘L) = the transformation defined by the ith iteration of the decryption

algorithm for 1 … I … 16 T17(R ‘L) = L ‘R, where this transformation occurs after the sixteenth iteration

of the encryption algorithm

a. Show that the composition TD1(IP(IP -1(T17(T16(L15 ‘R15))))) is equivalent to the

transformation that interchanges the 32-bit halves, L15 and R15. That is, show that

TD1(IP(IP -1(T17(T16(L15 ‘R15))))) = R15 ‘L15

b. Now suppose that we did away with the final 32-bit swap in the encryption algo- rithm. Then we would want the following equality to hold:

TD1(IP(IP -1(T16(L15 ‘R15)))) = L15 ‘R15

Does it?

Note: The following problems refer to details of DES that are described in Appendix S.

4.9 Consider the substitution defined by row 1 of S-box S1 in Table S.2. Show a block diagram similar to Figure 4.2 that corresponds to this substitution.

4.10 Compute the bits number 4, 17, 41, and 45 at the output of the first round of the DES decryption, assuming that the ciphertext block is composed of all ones and the exter- nal key is composed of all ones.

4.11 This problem provides a numerical example of encryption using a one-round version of DES. We start with the same bit pattern for the key K and the plaintext, namely:

Hexadecimal notation: 0 1 2 3 4 5 6 7 8 9 A B C D E F

Binary notation: 0000 0001 0010 0011 0100 0101 0110 0111

1000 1001 1010 1011 1100 1101 1110 1111

a. Derive K1, the first-round subkey. b. Derive L0, R0. c. Expand R0 to get E[R0], where E[ # ] is the expansion function of Table S.1. d. Calculate A = E[R0]⊕ K1. e. Group the 48-bit result of (d) into sets of 6 bits and evaluate the corresponding

S-box substitutions. f. Concatenate the results of (e) to get a 32-bit result, B.

140 CHAPTER 4 / BLOCK CIPHERS AND THE DATA ENCRYPTION STANDARD

g. Apply the permutation to get P(B). h. Calculate R1 = P(B)⊕ L0. i. Write down the ciphertext.

4.12 Analyze the amount of left shifts in the DES key schedule by studying Table S.3 (d). Is there a pattern? What could be the reason for the choice of these constants?

4.13 When using the DES algorithm for decryption, the 16 keys (K1, K2, c , K16) are used in reverse order. Therefore, the right-hand side of Figure S.1 is not valid for decryption. Design a key-generation scheme with the appropriate shift schedule (analogous to Table S.3d) for the decryption process.

4.14 a. Let X′ be the bitwise complement of X. Prove that if the complement of the plaintext block is taken and the complement of an encryption key is taken, then the result of DES encryption with these values is the complement of the original ciphertext. That is,

If Y = E(K, X)

Then Y′ = E(K′, X′)

Hint: Begin by showing that for any two bit strings of equal length, A and B, (A⊕ B)′ = A′ ⊕ B.

b. It has been said that a brute-force attack on DES requires searching a key space of 256 keys. Does the result of part (a) change that?

4.15 a. We say that a DES key K is weak if DESK is an involution. Exhibit four weak keys for DES.

b. We say that a DES key K is semi-weak if it is not weak and if there exists a key K′ such that DESK

– 1 = DESK′. Exhibit four semi-weak keys for DES.

Note: The following problems refer to simplified DES, described in Appendix G. 4.16 Refer to Figure G.3, which explains encryption function for S-DES.

a. How important is the initial permutation IP? b. How important is the SW function in the middle?

4.17 The equations for the variables q and r for S-DES are defined in the section on S-DES analysis. Provide the equations for s and t.

4.18 Using S-DES, decrypt the string 01000110 using the key 1010000010 by hand. Show intermediate results after each function (IP, FK, SW, FK, IP

-1). Then decode the first 4 bits of the plaintext string to a letter and the second 4 bits to another letter where we encode A through P in base 2 (i.e., A = 0000, B = 0001, c , P = 1111). Hint: As a midway check, after the xoring with K2, the string should be 11000001.

Programming Problems

4.19 Create software that can encrypt and decrypt using a general substitution block cipher.

4.20 Create software that can encrypt and decrypt using S-DES. Test data: use plaintext, ciphertext, and key of Problem 4.18.

141

5.1 Groups

Groups Abelian Group Cyclic Group

5.2 Rings

5.3 Fields

5.4 Finite Fields of the Form GF(p)

Finite Fields of Order p Finding the Multiplicative Inverse in GF(p) Summary

5.5 Polynomial Arithmetic

Ordinary Polynomial Arithmetic Polynomial Arithmetic with Coefficients in Zp Finding the Greatest Common Divisor Summary

5.6 Finite Fields of the form GF(2n)

Motivation Modular Polynomial Arithmetic Finding the Multiplicative Inverse Computational Considerations Using a Generator Summary

5.7 Key Terms, Review Questions, and Problems

CHAPTER

Finite Fields

142 CHAPTER 5 / FINITE FIELDS

Finite fields have become increasingly important in cryptography. A number of cryptographic algorithms rely heavily on properties of finite fields, notably the Advanced Encryption Standard (AES) and elliptic curve cryptography. Other exam- ples include the message authentication code CMAC and the authenticated encryption scheme GCM.

This chapter provides the reader with sufficient background on the concepts of finite fields to be able to understand the design of AES and other cryptographic algo- rithms that use finite fields. Because students unfamiliar with abstract algebra may find the concepts behind finite fields somewhat difficult to grasp, we approach the topic in a way designed to enhance understanding. Our plan of attack is as follows:

1. Fields are a subset of a larger class of algebraic structures called rings, which are in turn a subset of the larger class of groups. In fact, as shown in Figure 5.1, both groups and rings can be further differentiated. Groups are defined by a simple set of properties and are easily understood. Each successive subset (abelian group, ring, commutative ring, and so on) adds additional properties and is thus more complex. Sections 5.1 through 5.3 will examine groups, rings, and fields, successively.

2. Finite fields are a subset of fields, consisting of those fields with a finite num- ber of elements. These are the class of fields that are found in cryptographic algorithms. With the concepts of fields in hand, we turn in Section 5.4 to a specific class of finite fields, namely those with p elements, where p is prime. Certain asymmetric cryptographic algorithms make use of such fields.

3. A more important class of finite fields, for cryptography, comprises those with 2n elements depicted as fields of the form GF(2n). These are used in a wide variety of cryptographic algorithms. However, before discussing these fields, we need to analyze the topic of polynomial arithmetic, which is done in Section 5.5.

4. With all of this preliminary work done, we are able at last, in Section 5.6, to discuss finite fields of the form GF(2n).

Before proceeding, the reader may wish to review Sections 2.1 through 2.3, which cover relevant topics in number theory.

LEARNING OBJECTIVES

After studying this chapter, you should be able to:

◆ Distinguish among groups, rings, and fields.

◆ Define finite fields of the form GF(p).

◆ Explain the differences among ordinary polynomial arithmetic, polynomial arithmetic with coefficients in Zp, and modular polynomial arithmetic in GF(2n).

◆ Define finite fields of the form GF(2n).

◆ Explain the two different uses of the mod operator.

5.1 / GROUPS 143

5.1 GROUPS

Groups, rings, and fields are the fundamental elements of a branch of mathematics known as abstract algebra, or modern algebra. In abstract algebra, we are concerned with sets on whose elements we can operate algebraically; that is, we can combine two elements of the set, perhaps in several ways, to obtain a third element of the set. These operations are subject to specific rules, which define the nature of the set. By convention, the notation for the two principal classes of operations on set elements is usually the same as the notation for addition and multiplication on ordinary numbers. However, it is important to note that, in abstract algebra, we are not limited to ordi- nary arithmetical operations. All this should become clear as we proceed.

Groups

A group G, sometimes denoted by {G, # }, is a set of elements with a binary opera- tion denoted by # that associates to each ordered pair (a, b) of elements in G an element (a # b) in G, such that the following axioms are obeyed:1

(A1) Closure: If a and b belong to G, then a # b is also in G. (A2) Associative: a # (b # c) = (a # b) # c for all a, b, c in G.

1 The operator # is generic and can refer to addition, multiplication, or some other mathematical operation.

Figure 5.1 Groups, Rings, and Fields

Groups

Abelian groups

Rings

Commutative rings

Integral domains

Fields

Finite fields

144 CHAPTER 5 / FINITE FIELDS

(A3) Identity element: There is an element e in G such that a # e = e # a = a for all a in G.

(A4) Inverse element: For each a in G, there is an element a′ in G such that a # a′ = a′ # a = e.

Let Nn denote a set of n distinct symbols that, for convenience, we represent as {1, 2, c , n}. A permutation of n distinct symbols is a one-to-one mapping from Nn to Nn.

2 Define Sn to be the set of all permutations of n distinct symbols. Each element of Sn is represented by a permutation p of the integers in 1, 2, . . . , n. It is easy to demonstrate that Sn is a group:

A1: If (p, r∈ Sn), then the composite mapping p # r is formed by per- muting the elements of r according to the permutation p. For example, {3, 2, 1} # {1, 3, 2} = {2, 3, 1}. The notation for this map- ping is explained as follows: The value of the first element of p indicates which element of r is to be in the first position in p # r; the value of the second element of p indicates which element of r is to be in the second position in p # r; and so on. Clearly, p # r∈ Sn.

A2: The composition of mappings is also easily seen to be associative.

A3: The identity mapping is the permutation that does not alter the order of the n elements. For Sn, the identity element is {1, 2, c , n}.

A4: For any p∈ Sn, the mapping that undoes the permutation defined by p is the inverse element for p. There will always be such an inverse. For example {2, 3, 1} # {3, 1, 2} = {1, 2, 3}.

2This is equivalent to the definition of permutation in Chapter 2, which stated that a permutation of a finite set of elements S is an ordered sequence of all the elements of S, with each element appearing exactly once.

The set of integers (positive, negative, and 0) under addition is an abelian group. The set of nonzero real numbers under multiplication is an abelian group. The set Sn from the preceding example is a group but not an abelian group for n 7 2.

If a group has a finite number of elements, it is referred to as a finite group, and the order of the group is equal to the number of elements in the group. Otherwise, the group is an infinite group.

Abelian Group

A group is said to be abelian if it satisfies the following additional condition:

(A5) Commutative: a # b = b # a for all a, b in G.

5.2 / RINGS 145

When the group operation is addition, the identity element is 0; the in- verse element of a is -a; and subtraction is defined with the following rule: a – b = a + (-b).

Cyclic Group

We define exponentiation within a group as a repeated application of the group operator, so that a3 = a # a # a. Furthermore, we define a0 = e as the identity ele- ment, and a-n = (a′)n, where a′ is the inverse element of a within the group. A group G is cyclic if every element of G is a power ak (k is an integer) of a fixed element a∈G. The element a is said to generate the group G or to be a generator of G. A cyclic group is always abelian and may be finite or infinite.

The additive group of integers is an infinite cyclic group generated by the element 1. In this case, powers are interpreted additively, so that n is the nth power of 1.

5.2 RINGS

A ring R, sometimes denoted by {R, + , * }, is a set of elements with two binary operations, called addition and multiplication,3 such that for all a, b, c in R the fol- lowing axioms are obeyed.

(A1–A5) R is an abelian group with respect to addition; that is, R satisfies axioms A1 through A5. For the case of an additive group, we denote the identity element as 0 and the inverse of a as -a. (M1) Closure under multiplication: If a and b belong to R, then ab is also in R.

(M2) Associativity of multiplication: a(bc) = (ab)c for all a, b, c in R.

(M3) Distributive laws: a(b + c) = ab + ac for all a, b, c in R. (a + b)c = ac + bc for all a, b, c in R.

In essence, a ring is a set of elements in which we can do addition, subtraction [a – b = a + (-b)], and multiplication without leaving the set.

3Generally, we do not use the multiplication symbol, * , but denote multiplication by the concatenation of two elements.

With respect to addition and multiplication, the set of all n-square matrices over the real numbers is a ring.

A ring is said to be commutative if it satisfies the following additional condition:

(M4) Commutativity of multiplication: ab = ba for all a, b in R.

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146 CHAPTER 5 / FINITE FIELDS

Next, we define an integral domain, which is a commutative ring that obeys the following axioms.

(M5) Multiplicative identity: There is an element 1 in R such that a1 = 1a = a for all a in R.

(M6) No zero divisors: If a, b in R and ab = 0, then either a = 0 or b = 0.

Let S be the set of even integers (positive, negative, and 0) under the usual operations of addition and multiplication. S is a commutative ring. The set of all n-square matrices defined in the preceding example is not a commutative ring.

The set Zn of integers {0, 1, c , n – 1}, together with the arithmetic oper- ations modulo n, is a commutative ring (Table 4.3).

Let S be the set of integers (positive, negative, and 0) under the usual operations of addition and multiplication. S is an integral domain.

Familiar examples of fields are the rational numbers, the real numbers, and the complex numbers. Note that the set of all integers is not a field, because not every element of the set has a multiplicative inverse; in fact, only the elements 1 and -1 have multiplicative inverses in the integers.

5.3 FIELDS

A field F, sometimes denoted by {F, + , * }, is a set of elements with two binary operations, called addition and multiplication, such that for all a, b, c in F the follow- ing axioms are obeyed.

(A1–M6) F is an integral domain; that is, F satisfies axioms A1 through A5 and M1 through M6.

(M7) Multiplicative inverse: For each a in F, except 0, there is an element a-1 in F such that aa-1 = (a-1)a = 1.

In essence, a field is a set of elements in which we can do addition, subtraction, multiplication, and division without leaving the set. Division is defined with the fol- lowing rule: a/b = a(b-1).

In gaining insight into fields, the following alternate characterization may be useful. A field F, denoted by {F, +}, is a set of elements with two binary operations, called addition and multiplication, such that the following conditions hold:

1. F forms an abelian group with respect to addition.

2. The nonzero elements of F form an abelian group with respect to multiplication.

5.4 / FINITE FIELDS OF THE FORM GF(p) 147

3. The distributive law holds. That is, for all a, b, c in F,

a(b + c) = ab + ac.

(a + b)c = ac + bc

4. Figure 5.2 summarizes the axioms that define groups, rings, and fields.

5.4 FINITE FIELDS OF THE FORM GF(p)

In Section 5.3, we defined a field as a set that obeys all of the axioms of Figure 5.2 and gave some examples of infinite fields. Infinite fields are not of particular inter- est in the context of cryptography. However, in addition to infinite fields, there are two types of finite fields, as illustrated in Figure 5.3. Finite fields play a crucial role in many cryptographic algorithms.

It can be shown that the order of a finite field (number of elements in the field) must be a power of a prime pn, where n is a positive integer. The finite field of order pn is generally written GF(pn); GF stands for Galois field, in honor of the mathematician who first studied finite fields. Two special cases are of interest for our purposes. For n = 1, we have the finite field GF(p); this finite field has a differ- ent structure than that for finite fields with n 7 1 and is studied in this section. For finite fields of the form GF(pn), GF(2n) fields are of particular cryptographic inter- est, and these are covered in Section 5.6.

Finite Fields of Order p

For a given prime, p, we define the finite field of order p, GF(p), as the set Zp of integers {0, 1, c , p – 1} together with the arithmetic operations modulo p. Note therefore that we are using ordinary modular arithmetic to define the operations over these fields.

Figure 5.2 Properties of Groups, Rings, and Fields

(A1) Closure under addition: If a and b belong to S, then a + b is also in S (A2) Associativity of addition: a + (b + c) = (a + b) + c for all a, b, c in S (A3) Additive identity: There is an element 0 in R such that

a + 0 = 0 + a = a for all a in S (A4) Additive inverse: For each a in S there is an element –a in S

such that a + (–a) = (–a) + a = 0

(A5) Commutativity of addition: a + b = b + a for all a, b in S

(M1) Closure under multiplication: If a and b belong to S, then ab is also in S (M2) Associativity of multiplication: a(bc) = (ab)c for all a, b, c in S (M3) Distributive laws: a(b + c) = ab + ac for all a, b, c in S

(a + b)c = ac + bc for all a, b, c in S

(M4) Commutativity of multiplication: ab = ba for all a, b in S

(M5) Multiplicative identity: There is an element 1 in S such that a1 = 1a = a for all a in S

(M6) No zero divisors: If a, b in S and ab = 0, then either a = 0 or b = 0

(M7) Multiplicative inverse: If a belongs to S and a ≠ 0, there is an element a –1 in S such that aa –1 = a –1a = 1

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148 CHAPTER 5 / FINITE FIELDS

Recall that we showed in Section 5.2 that the set Zn of integers {0, 1, c , n – 1}, together with the arithmetic operations modulo n, is a commutative ring (Table 2.5). We further observed that any integer in Zn has a multiplicative inverse if and only if that integer is relatively prime to n [see discussion of Equation (2.5)].4 If n is prime, then all of the nonzero integers in Zn are relatively prime to n, and therefore there exists a multiplicative inverse for all of the nonzero integers in Zn. Thus, for Zp we can add the following properties to those listed in Table 5.2:

Multiplicative inverse (w-1)

For each w ∈ Zp, w ≠ 0, there exists a z∈ Zp such that w * z K 1 (mod p)

Because w is relatively prime to p, if we multiply all the elements of Zp by w, the resulting residues are all of the elements of Zp permuted. Thus, exactly one of the residues has the value 1. Therefore, there is some integer in Zp that, when multiplied by w, yields the residue 1. That integer is the multiplicative inverse of w, designated w-1. Therefore, Zp is in fact a finite field. Furthermore, Equation (2.5) is consistent with the existence of a multiplicative inverse and can be rewritten with- out the condition:

if (a * b) K (a * c)(mod p) then b K c(mod p) (5.1)

Multiplying both sides of Equation (5.1) by the multiplicative inverse of a, we have

((a-1) * a * b) K ((a-1) * a * c)(mod p) b K c (mod p)

4As stated in the discussion of Equation (2.5), two integers are relatively prime if their only common positive integer factor is 1.

Figure 5.3 Types of Fields

Fields

Fields with an infinite number

of elements

Finite fields

GF(p) Finite fields

with p elements

GF(pn) Finite fields

with pn elements

The simplest finite field is GF(2). Its arithmetic operations are easily summarized:

+ 0 1 0 0 1

1 1 0

Addition

* 0 1 0 0 0

1 0 1

Multiplication

w -w w-1

0 0 – 1 1 1

Inverses

In this case, addition is equivalent to the exclusive-OR (XOR) operation, and multiplication is equivalent to the logical AND operation.

5.4 / FINITE FIELDS OF THE FORM GF(p) 149

The right-hand side of Table 5.1 shows arithmetic operations in GF(7). This is a field of order 7 using modular arithmetic modulo 7. As can be seen, it satisfies all of the properties required of a field (Figure 5.2). Compare with the left-hand side of Table 5.1, which reproduces Table 2.2. In the latter case, we see that the set Z8, using modular arithmetic modulo 8, is not a field. Later in this chapter, we show how to define addition and multiplication operations on Z8 in such a way as to form a finite field.

Finding the Multiplicative Inverse in GF(p)

It is easy to find the multiplicative inverse of an element in GF(p) for small values of p. You simply construct a multiplication table, such as shown in Table 5.1e, and the desired result can be read directly. However, for large values of p, this approach is not practical.

If a and b are relatively prime, then b has a multiplicative inverse modulo a. That is, if gcd(a, b) = 1, then b has a multiplicative inverse modulo a. That is, for positive integer b 6 a, there exists a b-1 6 a such that bb-1 = 1 mod a. If a is a prime number and b 6 a, then clearly a and b are relatively prime and have a great- est common divisor of 1. We now show that we can easily compute b-1 using the extended Euclidean algorithm.

We repeat here Equation (2.7), which we showed can be solved with the ex- tended Euclidean algorithm:

ax + by = d = gcd(a, b)

Now, if gcd(a, b) = 1, then we have ax + by = 1. Using the basic equalities of modular arithmetic, defined in Section 2.3, we can say

[(ax mod a) + (by mod a)] mod a = 1 mod a 0 + (by mod a) = 1

But if by mod a = 1, then y = b-1. Thus, applying the extended Euclidean algorithm to Equation (2.7) yields the value of the multiplicative inverse of b if gcd(a, b) = 1.

Consider the example that was shown in Table 2.4. Here we have a = 1759, which is a prime number, and b = 550. The solution of the equation 1759x + 550y = d yields a value of y = 355. Thus, b-1 = 355. To verify, we cal- culate 550 * 355 mod 1759 = 195250 mod 1759 = 1.

More generally, the extended Euclidean algorithm can be used to find a multiplicative inverse in Zn for any n. If we apply the extended Euclidean algorithm to the equation nx + by = d, and the algorithm yields d = 1, then y = b-1 in Zn.

150 CHAPTER 5 / FINITE FIELDS

+ 0 1 2 3 4 5 6 7

0 0 1 2 3 4 5 6 7

1 1 2 3 4 5 6 7 0

2 2 3 4 5 6 7 0 1

3 3 4 5 6 7 0 1 2

4 4 5 6 7 0 1 2 3

5 5 6 7 0 1 2 3 4

6 6 7 0 1 2 3 4 5

7 7 0 1 2 3 4 5 6

(a) Addition modulo 8

* 0 1 2 3 4 5 6 7

0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7

2 0 2 4 6 0 2 4 6

3 0 3 6 1 4 7 2 5

4 0 4 0 4 0 4 0 4

5 0 5 2 7 4 1 6 3

6 0 6 4 2 0 6 4 2

7 0 7 6 5 4 3 2 1

(b) Multiplication modulo 8

w 0 1 2 3 4 5 6 7

-w 0 7 6 5 4 3 2 1

w-1 — 1 — 3 — 5 — 7

(c) Additive and multiplicative inverses modulo 8

+ 0 1 2 3 4 5 6

0 0 1 2 3 4 5 6

1 1 2 3 4 5 6 0

2 2 3 4 5 6 0 1

3 3 4 5 6 0 1 2

4 4 5 6 0 1 2 3

5 5 6 0 1 2 3 4

6 6 0 1 2 3 4 5

(d) Addition modulo 7

* 0 1 2 3 4 5 6

0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6

2 0 2 4 6 1 3 5

3 0 3 6 2 5 1 4

4 0 4 1 5 2 6 3

5 0 5 3 1 6 4 2

6 0 6 5 4 3 2 1

(e) Multiplication modulo 7

w 0 1 2 3 4 5 6

-w 0 6 5 4 3 2 1

w-1 — 1 4 5 2 3 6

(f) Additive and multiplicative inverses modulo 7

Table 5.1 Arithmetic Modulo 8 and Modulo 7

Summary

In this section, we have shown how to construct a finite field of order p, where p is prime. Specifically, we defined GF(p) with the following properties.

1. GF(p) consists of p elements.

2. The binary operations + and * are defined over the set. The operations of addition, subtraction, multiplication, and division can be performed without leaving the set. Each element of the set other than 0 has a multiplicative in- verse, and division is performed by multiplication by the multiplicative inverse.

We have shown that the elements of GF(p) are the integers {0, 1, c , p – 1} and that the arithmetic operations are addition and multiplication mod p.

5.5 / POLYNOMIAL ARITHMETIC 151

5.5 POLYNOMIAL ARITHMETIC

Before continuing our discussion of finite fields, we need to introduce the interest- ing subject of polynomial arithmetic. We are concerned with polynomials in a single variable x, and we can distinguish three classes of polynomial arithmetic (Figure 5.4).

■ Ordinary polynomial arithmetic, using the basic rules of algebra.

■ Polynomial arithmetic in which the arithmetic on the coefficients is performed modulo p; that is, the coefficients are in GF(p).

■ Polynomial arithmetic in which the coefficients are in GF(p), and the poly- nomials are defined modulo a polynomial m(x) whose highest power is some integer n.

This section examines the first two classes, and the next section covers the last class.

Ordinary Polynomial Arithmetic

A polynomial of degree n (integer n Ú 0) is an expression of the form

f(x) = anxn + an – 1xn – 1 + g + a1x + a0 = a n

i=0 aix

i

where the ai are elements of some designated set of numbers S, called the coefficient set, and an ≠ 0. We say that such polynomials are defined over the coefficient set S.

A zero-degree polynomial is called a constant polynomial and is simply an element of the set of coefficients. An nth-degree polynomial is said to be a monic polynomial if an = 1.

In the context of abstract algebra, we are usually not interested in evaluating a polynomial for a particular value of x [e.g., f(7)]. To emphasize this point, the vari- able x is sometimes referred to as the indeterminate.

Polynomial arithmetic includes the operations of addition, subtraction, and multiplication. These operations are defined in a natural way as though the variable

Figure 5.4 Treatment of Polynomials

Polynomial f(x)

x treated as a variable, and evaluated for

a particular value of x

x treated as an indeterminate

Ordinary polynomial arithmetic

Arithmetic on coefficients is

performed modulo p

Arithmetic on coefficients is performed modulo p

and polynomials are defined modulo a polynomial m(x)

152 CHAPTER 5 / FINITE FIELDS

x was an element of S. Division is similarly defined, but requires that S be a field. Examples of fields include the real numbers, rational numbers, and Zp for p prime. Note that the set of all integers is not a field and does not support polynomial division.

Addition and subtraction are performed by adding or subtracting correspond- ing coefficients. Thus, if

f(x) = a n

i=0 aix

i; g(x) = a m

i=0 bix

i; n Ú m

then addition is defined as

f(x) + g(x) = a m

i=0 (ai + bi)xi + a

n

i=m + 1 aix

i

and multiplication is defined as

f(x) * g(x) = a n + m

i=0 cix

i

where

ck = a0bk + a1bk – 1 + g + ak – 1b1 + akb0

In the last formula, we treat ai as zero for i 7 n and bi as zero for i 7 m. Note that the degree of the product is equal to the sum of the degrees of the two polynomials.

As an example, let f(x) = x3 + x2 + 2 and g(x) = x2 – x + 1, where S is the set of integers. Then

f(x) + g(x) = x3 + 2×2 – x + 3 f(x) – g(x) = x3 + x + 1 f(x) * g(x) = x5 + 3×2 – 2x + 2

Figures 5.5a through 5.5c show the manual calculations. We comment on division subsequently.

Polynomial Arithmetic with Coefficients in Zp Let us now consider polynomials in which the coefficients are elements of some field F; we refer to this as a polynomial over the field F. In this case, it is easy to show that the set of such polynomials is a ring, referred to as a polynomial ring. That is, if we consider each distinct polynomial to be an element of the set, then that set is a ring.5

When polynomial arithmetic is performed on polynomials over a field, then division is possible. Note that this does not mean that exact division is possible. Let

5In fact, the set of polynomials whose coefficients are elements of a commutative ring forms a polynomial ring, but that is of no interest in the present context.

5.5 / POLYNOMIAL ARITHMETIC 153

us clarify this distinction. Within a field, given two elements a and b, the quotient a/b is also an element of the field. However, given a ring R that is not a field, in gen- eral, division will result in both a quotient and a remainder; this is not exact division.

Figure 5.5 Examples of Polynomial Arithmetic

x3

x3

+ +x2

+2×2 x2 x

2

+–+ ( )

× ( )

– ( )

x–

1

+ 3

(a) Addition

(d) Division(c) Multiplication

x3

x3

+ +x2

+ x2 x2 x

2

x3 x 2

+

+

+x2

x3 – x2

2×2 + x

– x

x

2

+ 2

2×2– 2x + 2

x4 –– –x3 2x

– 2x

x5 + +x4 2×2

x5 +3×2

+– 1 x2 x +– 1

+ 2

+ 2

x3

x3

+ +x2

x2 x

2

+–

x+

1

+ 1

(b) Subtraction

Consider the division 5/3 within a set S. If S is the set of rational numbers, which is a field, then the result is simply expressed as 5/3 and is an element of S. Now suppose that S is the field Z7. In this case, we calculate (using Table 5.1f)

5/3 = (5 * 3-1) mod 7 = (5 * 5) mod 7 = 4 which is an exact solution. Finally, suppose that S is the set of integers, which is a ring but not a field. Then 5/3 produces a quotient of 1 and a remainder of 2:

5/3 = 1 + 2/3 5 = 1 * 3 + 2

Thus, division is not exact over the set of integers.

Now, if we attempt to perform polynomial division over a coefficient set that is not a field, we find that division is not always defined.

If the coefficient set is the integers, then (5×2)/(3x) does not have a solution, because it would require a coefficient with a value of 5/3, which is not in the coef- ficient set. Suppose that we perform the same polynomial division over Z7. Then we have (5×2)/(3x) = 4x, which is a valid polynomial over Z7.

However, as we demonstrate presently, even if the coefficient set is a field, polynomial division is not necessarily exact. In general, division will produce a quo- tient and a remainder. We can restate the division algorithm of Equation (2.1) for polynomials over a field as follows. Given polynomials f(x) of degree n and g(x)

154 CHAPTER 5 / FINITE FIELDS

of degree (m), (n Ú m), if we divide f(x) by g(x), we get a quotient q(x) and a remainder r(x) that obey the relationship

f(x) = q(x)g(x) + r(x) (5.2)

with polynomial degrees:

Degree f(x) = n Degree g(x) = m Degree q(x) = n – m Degree r(x) … m – 1

With the understanding that remainders are allowed, we can say that poly- nomial division is possible if the coefficient set is a field. One common technique used for polynomial division is polynomial long division, similar to long division for integers. Examples of this are shown subsequently.

In an analogy to integer arithmetic, we can write f(x) mod g(x) for the remain- der r(x) in Equation (5.2). That is, r(x) = f(x) mod g(x). If there is no remainder [i.e., r(x) = 0], then we can say g(x) divides f(x), written as g(x) � f(x). Equivalently, we can say that g(x) is a factor of f(x) or g(x) is a divisor of f(x).

For the preceding example [f(x) = x3 + x2 + 2 and g(x) = x2 – x + 1], f(x)/g(x) produces a quotient of q(x) = x + 2 and a remainder r(x) = x, as shown in Figure 5.5d. This is easily verified by noting that

q(x)g(x) + r(x) = (x + 2)(x2 – x + 1) + x = (x3 + x2 – x + 2) + x = x3 + x2 + 2 = f(x)

For our purposes, polynomials over GF(2) are of most interest. Recall from Section 5.4 that in GF(2), addition is equivalent to the XOR operation, and multi- plication is equivalent to the logical AND operation. Further, addition and subtrac- tion are equivalent mod 2:

1 + 1 = 1 – 1 = 0 1 + 0 = 1 – 0 = 1 0 + 1 = 0 – 1 = 1

Figure 5.6 shows an example of polynomial arithmetic over GF(2). For f(x) = (x7 + x5 + x4 + x3 + x + 1) and g(x) = (x3 + x + 1), the figure shows f(x) + g(x); f(x) – g(x); f(x) * g(x); and f(x)/g(x). Note that g(x) � f(x).

A polynomial f(x) over a field F is called irreducible if and only if f(x) can- not be expressed as a product of two polynomials, both over F, and both of degree lower than that of f(x). By analogy to integers, an irreducible polynomial is also called a prime polynomial.

The polynomial6 f(x) = x4 + 1 over GF(2) is reducible, because x4 + 1 = (x + 1)(x3 + x2 + x + 1).

6In the reminder of this chapter, unless otherwise noted, all examples are of polynomials over GF(2).

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5.5 / POLYNOMIAL ARITHMETIC 155

Consider the polynomial f(x) = x3 + x + 1. It is clear by inspection that x is not a factor of f(x). We easily show that x + 1 is not a factor of f(x):

x2 + x x + 1�x3 + x + 1

x3 + x2

x2 + x x2 + x

1 Thus, f(x) has no factors of degree 1. But it is clear by inspection that if f(x) is reducible, it must have one factor of degree 2 and one factor of degree 1. There- fore, f(x) is irreducible.

Figure 5.6 Examples of Polynomial Arithmetic over GF(2)

(a) Addition

(c) Multiplication

(d) Division

x4x5 ++x7 xx3

x3x4 ++x5 ++x7 +x 1

+++ ( )1

x3x4 ++x5 ++x7 +x 1

x4x5 ++x7 x3 x

x3 ++ +x 1

+ 1

x5x6 ++x8 x4 ++ +x2

+ x2

x

x7x8 ++x10 x6 ++ +x4

x10 + x4 x3

++× ( )1

x3x4 ++x5 ++x7

x4x5 ++x7

+x

x3 x

1

++– ( )1

(b) Subtraction

x3x4 ++x5 ++

++

x7

x4x5x7 +x 1

x3 + +x 1

x3 + +x 1

x4 1+

x3 x ++ 1

156 CHAPTER 5 / FINITE FIELDS

Finding the Greatest Common Divisor

We can extend the analogy between polynomial arithmetic over a field and integer arithmetic by defining the greatest common divisor as follows. The polynomial c(x) is said to be the greatest common divisor of a(x) and b(x) if the following are true.

1. c(x) divides both a(x) and b(x).

2. Any divisor of a(x) and b(x) is a divisor of c(x).

An equivalent definition is the following: gcd[a(x), b(x)] is the polynomial of maximum degree that divides both a(x) and b(x).

We can adapt the Euclidean algorithm to compute the greatest common divisor of two polynomials. Recall Equation (2.6), from Chapter 2, which is the basis of the Euclidean algorithm: gcd(a, b) = gcd(b, a mod b). This equality can be rewritten as the following equation:

gcd[a(x), b(x)] = gcd[b(x), a(x) mod b(x)] (5.3)

Equation (5.3) can be used repetitively to determine the greatest common divisor. Compare the following scheme to the definition of the Euclidean algorithm for integers.

Euclidean Algorithm for Polynomials

Calculate Which satisfies

r1(x) = a(x) mod b(x) a(x) = q1(x)b(x) + r1(x) r2(x) = b(x) mod r1(x) b(x) = q2(x)r1(x) + r2(x) r3(x) = r1(x) mod r2(x) r1(x) = q3(x)r2(x) + r3(x)

rn(x) = rn – 2(x) mod rn – 1(x) rn – 2(x) = qn(x)rn – 1(x) + rn(x)

rn + 1(x) = rn – 1(x) mod rn(x) = 0 rn – 1(x) = qn + 1(x)rn(x) + 0

d(x) = gcd(a(x), b(x)) = rn(x)

At each iteration, we have d(x) = gcd(ri+ 1(x), ri(x)) until finally d(x) = gcd(rn(x), 0) = rn(x). Thus, we can find the greatest common divisor of two integers by repetitive application of the division algorithm. This is the Euclidean algorithm for polynomials. The algorithm assumes that the degree of a(x) is greater than the degree of b(x).

Find gcd[a(x), b(x)] for a(x) = x6 + x5 + x4 + x3 + x2 + x + 1 and b(x) = x4 + x2 + x + 1. First, we divide a(x) by b(x):

x2 + x x4 + x2 + x + 1�x6 + x5 + x4 + x3 + x2 + x + 1

x6 + x4 + x3 + x2

x5 + x + 1 x5 + x3 + x2 + x

x3 + x2 + 1

5.6 / FINITE FIELDS OF THE FORM GF(2n) 157

Summary

We began this section with a discussion of arithmetic with ordinary polynomials. In ordinary polynomial arithmetic, the variable is not evaluated; that is, we do not plug a value in for the variable of the polynomials. Instead, arithmetic operations are performed on polynomials (addition, subtraction, multiplication, division) using the ordinary rules of algebra. Polynomial division is not allowed unless the coefficients are elements of a field.

Next, we discussed polynomial arithmetic in which the coefficients are ele- ments of GF(p). In this case, polynomial addition, subtraction, multiplication, and division are allowed. However, division is not exact; that is, in general division re- sults in a quotient and a remainder.

Finally, we showed that the Euclidean algorithm can be extended to find the greatest common divisor of two polynomials whose coefficients are elements of a field.

All of the material in this section provides a foundation for the following sec- tion, in which polynomials are used to define finite fields of order pn.

5.6 FINITE FIELDS OF THE FORM GF(2n)

Earlier in this chapter, we mentioned that the order of a finite field must be of the form pn, where p is a prime and n is a positive integer. In Section 5.4, we looked at the special case of finite fields with order p. We found that, using modular arith- metic in Zp, all of the axioms for a field (Figure 5.2) are satisfied. For polynomials over pn, with n 7 1, operations modulo pn do not produce a field. In this section, we show what structure satisfies the axioms for a field in a set with pn elements and concentrate on GF(2n).

Motivation

Virtually all encryption algorithms, both symmetric and asymmetric, involve arith- metic operations on integers. If one of the operations that is used in the algorithm is division, then we need to work in arithmetic defined over a field. For convenience

This yields r1(x) = x3 + x2 + 1 and q1 (x) = x2 + x. Then, we divide b(x) by r1(x).

x + 1 x3 + x2 + 1�x4 + x2 + x + 1

x4 + x3 + x x3 + x2 + 1 x3 + x2 + 1

This yields r2(x) = 0 and q2(x) = x + 1. Therefore, gcd[a(x), b(x)] = r1(x) = x3 + x2 + 1.

158 CHAPTER 5 / FINITE FIELDS

and for implementation efficiency, we would also like to work with integers that fit exactly into a given number of bits with no wasted bit patterns. That is, we wish to work with integers in the range 0 through 2n – 1, which fit into an n-bit word.

Suppose we wish to define a conventional encryption algorithm that operates on data 8 bits at a time, and we wish to perform division. With 8 bits, we can repre- sent integers in the range 0 through 255. However, 256 is not a prime number, so that if arithmetic is performed in Z256 (arithmetic modulo 256), this set of inte- gers will not be a field. The closest prime number less than 256 is 251. Thus, the set Z251, using arithmetic modulo 251, is a field. However, in this case the 8-bit patterns representing the integers 251 through 255 would not be used, resulting in inefficient use of storage.

As the preceding example points out, if all arithmetic operations are to be used and we wish to represent a full range of integers in n bits, then arithmetic modulo 2n will not work. Equivalently, the set of integers modulo 2n for n 7 1, is not a field. Furthermore, even if the encryption algorithm uses only addition and multiplication, but not division, the use of the set Z2n is questionable, as the follow- ing example illustrates.

Suppose we wish to use 3-bit blocks in our encryption algorithm and use only the operations of addition and multiplication. Then arithmetic modulo 8 is well defined, as shown in Table 5.1. However, note that in the multiplication table, the nonzero integers do not appear an equal number of times. For example, there are only four occurrences of 3, but twelve occurrences of 4. On the other hand, as was mentioned, there are finite fields of the form GF(2n), so there is in particular a finite field of order 23 = 8. Arithmetic for this field is shown in Table 5.2. In this case, the number of occurrences of the nonzero integers is uniform for multiplication. To summarize,

Integer 1 2 3 4 5 6 7 Occurrences in Z8 4 8 4 12 4 8 4 Occurrences in GF(23) 7 7 7 7 7 7 7

For the moment, let us set aside the question of how the matrices of Table 5.2 were constructed and instead make some observations.

1. The addition and multiplication tables are symmetric about the main diago- nal, in conformance to the commutative property of addition and multiplica- tion. This property is also exhibited in Table 5.1, which uses mod 8 arithmetic.

2. All the nonzero elements defined by Table 5.2 have a multiplicative inverse, unlike the case with Table 5.1.

3. The scheme defined by Table 5.2 satisfies all the requirements for a finite field. Thus, we can refer to this scheme as GF(23).

4. For convenience, we show the 3-bit assignment used for each of the elements of GF(23).

5.6 / FINITE FIELDS OF THE FORM GF(2n) 159

Intuitively, it would seem that an algorithm that maps the integers unevenly onto themselves might be cryptographically weaker than one that provides a uni- form mapping. That is, a cryptanalytic technique might be able to exploit the fact that some integers occur more frequently and some less frequently in the ciphertext. Thus, the finite fields of the form GF(2n) are attractive for cryptographic algorithms.

To summarize, we are looking for a set consisting of 2n elements, together with a definition of addition and multiplication over the set that define a field. We can assign a unique integer in the range 0 through 2n – 1 to each element of the set. Keep in mind that we will not use modular arithmetic, as we have seen that this does not result in a field. Instead, we will show how polynomial arithmetic provides a means for constructing the desired field.

Modular Polynomial Arithmetic

Consider the set S of all polynomials of degree n – 1 or less over the field Zp. Thus, each polynomial has the form

f(x) = an – 1xn – 1 + an – 2xn – 2 + g + a1x + a0 = a n – 1

i=0 aix

i

000 001 010 011 100 101 110 111

+ 0 1 2 3 4 5 6 7

000 0 0 1 2 3 4 5 6 7

001 1 1 0 3 2 5 4 7 6

010 2 2 3 0 1 6 7 4 5

011 3 3 2 1 0 7 6 5 4

100 4 4 5 6 7 0 1 2 3

101 5 5 4 7 6 1 0 3 2

110 6 6 7 4 5 2 3 0 1

111 7 7 6 5 4 3 2 1 0

(a) Addition

000 001 010 011 100 101 110 111

* 0 1 2 3 4 5 6 7

000 0 0 0 0 0 0 0 0 0

001 1 0 1 2 3 4 5 6 7

010 2 0 2 4 6 3 1 7 5

011 3 0 3 6 5 7 4 1 2

100 4 0 4 3 7 6 2 5 1

101 5 0 5 1 4 2 7 3 6

110 6 0 6 7 1 5 3 2 4

111 7 0 7 5 2 1 6 4 3

(b) Multiplication

w -w w-1

0 0 –

1 1 1

2 2 5

3 3 6

4 4 7

5 5 2

6 6 3

7 7 4

(c) Additive and multiplicative inverses

Table 5.2 Arithmetic in GF(23)

160 CHAPTER 5 / FINITE FIELDS

where each ai takes on a value in the set {0, 1, c , p – 1}. There are a total of pn different polynomials in S.

For p = 3 and n = 2, the 32 = 9 polynomials in the set are 0, 1, 2, x, x + 1, x + 2, 2x, 2x + 1, 2x + 2

For p = 2 and n = 3, the 23 = 8 polynomials in the set are 0, 1, x, x + 1, x2, x2 + 1, x2 + x, x2 + x + 1

With the appropriate definition of arithmetic operations, each such set S is a finite field. The definition consists of the following elements.

1. Arithmetic follows the ordinary rules of polynomial arithmetic using the basic rules of algebra, with the following two refinements.

2. Arithmetic on the coefficients is performed modulo p. That is, we use the rules of arithmetic for the finite field Zp.

3. If multiplication results in a polynomial of degree greater than n – 1, then the polynomial is reduced modulo some irreducible polynomial m(x) of degree n. That is, we divide by m(x) and keep the remainder. For a polynomial f(x), the remainder is expressed as r(x) = f(x) mod m(x).

The Advanced Encryption Standard (AES) uses arithmetic in the finite field GF(28), with the irreducible polynomial m(x) = x8 + x4 + x3 + x + 1. Consider the two polynomials f(x) = x6 + x4 + x2 + x + 1 and g(x) = x7 + x + 1. Then

f(x) + g(x) = x6 + x4 + x2 + x + 1 + x7 + x + 1 = x7 + x6 + x4 + x2

f(x) * g(x) = x13 + x11 + x9 + x8 + x7

+ x7 + x5 + x3 + x2 + x + x6 + x4 + x2 + x + 1

= x13 + x11 + x9 + x8 + x6 + x5 + x4 + x3 + 1

x5 + x3

x8 + x4 + x3 + x + 1 >x13 + x11 + x9 + x8 + x6 + x5 + x4 + x3 + 1 x13 + x9 + x8 + x6 + x5

x11 + x4 + x3

x11 + x7 + x6 + x4 + x3

x7 + x6 + 1

Therefore, f(x) * g(x) mod m(x) = x7 + x6 + 1.

5.6 / FINITE FIELDS OF THE FORM GF(2n) 161

As with ordinary modular arithmetic, we have the notion of a set of residues in modular polynomial arithmetic. The set of residues modulo m(x), an nth-degree polynomial, consists of pn elements. Each of these elements is represented by one of the pn polynomials of degree m 6 n.

The residue class [x + 1], (mod m(x)), consists of all polynomials a(x) such that a(x) K (x + 1)(mod m(x)). Equivalently, the residue class [x + 1] consists of all polynomials a(x) that satisfy the equality a(x) mod m(x) = x + 1.

It can be shown that the set of all polynomials modulo an irreducible nth- degree polynomial m(x) satisfies the axioms in Figure 5.2, and thus forms a finite field. Furthermore, all finite fields of a given order are isomorphic; that is, any two finite-field structures of a given order have the same structure, but the representa- tion or labels of the elements may be different.

To construct the finite field GF(23), we need to choose an irreducible poly- nomial of degree 3. There are only two such polynomials: (x3 + x2 + 1) and (x3 + x + 1). Using the latter, Table 5.3 shows the addition and multiplication tables for GF(23). Note that this set of tables has the identical structure to those of Table 5.2. Thus, we have succeeded in finding a way to define a field of order 23.

We can now read additions and multiplications from the table easily. For exam- ple, consider binary 100 + 010 = 110. This is equivalent to x2 + x. Also consider 100 * 010 = 011, which is equivalent to x2 * x = x3 and reduces to x + 1. That is, x3 mod (x3 + x + 1) = x + 1, which is equivalent to 011.

Finding the Multiplicative Inverse

Just as the Euclidean algorithm can be adapted to find the greatest common divisor of two polynomials, the extended Euclidean algorithm can be adapted to find the multiplicative inverse of a polynomial. Specifically, the algorithm will find the mul- tiplicative inverse of b(x) modulo a(x) if the degree of b(x) is less than the degree of a(x) and gcd[a(x), b(x)] = 1. If a(x) is an irreducible polynomial, then it has no fac- tor other than itself or 1, so that gcd[a(x), b(x)] = 1. The algorithm can be charac- terized in the same way as we did for the extended Euclidean algorithm for integers. Given polynomials a(x) and b(x) with the degree of a(x) greater than the degree of b(x), we wish to solve the following equation for the values v(x), w(x), and d(x), where d(x) = gcd[a(x), b(x)]:

a(x)v(x) + b(x)w(x) = d(x)

If d(x) = 1, then w(x) is the multiplicative inverse of b(x) modulo a(x). The calcula- tions are as follows.

162 CHAPTER 5 / FINITE FIELDS

00 0

00 1

01 0

01 1

10 0

10 1

11 0

11 1

+ 0

1 x

x +

1 x2

x2 +

1 x2

+ x

x2 +

x +

1

00 0

0 0

1 x

x +

1 x2

x2 +

1 x2

+ x

x2 +

x +

1

00 1

1 1

0 x

+ 1

x x2

+ 1

x2 x2

+ x

+ 1

x2 +

x

01 0

x x

x +

1 0

1 x2

+ x

x2 +

x +

1 x2

x2 +

1

01 1

x +

1 x

+ 1

x 1

0 x2

+ x

+ 1

x2 +

x x2

+ 1

x2

10 0

x2 x2

x2 +

1 x2

+ x

x2 +

x +

1 0

1 x

x +

1

10 1

x2 +

1 x2

+ 1

x2 x2

+ x

+ 1

x2 +

x 1

0 x

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x

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x2 +

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x2 +

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x2 +

1 x

x +

1 0

1

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1 x2

+ x

+ 1

x2 +

x x2

+ 1

x2 x

+ 1

x 1

0

00 0

00 1

01 0

01 1

10 0

10 1

11 0

11 1

* 0

1 x

x +

1 x2

x2 +

1 x2

+ x

x2 +

x +

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00 0

0 0

0 0

0 0

0 0

0

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1 x2

x2 +

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x x2

+ 1

1

10 1

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1 0

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x2 +

x +

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x2 +

x

11 0

x2 +

x 0

x2 +

x x2

+ x

+ 1

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+ 1

x +

1 x

x2

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5.6 / FINITE FIELDS OF THE FORM GF(2n) 163

Extended Euclidean Algorithm for Polynomials

Calculate Which satisfies Calculate Which satisfies

r-1(x) = a(x) v-1(x) = 1; w-1(x) = 0 a(x) = a(x)v-1(x) + bw-1(x)

r0(x) = b(x) v0(x) = 0; w0(x) = 1 b(x) = a(x)v0(x) + b(x)w0(x)

r1(x) = a(x) mod b(x) q1(x) = quotient of a(x)/b(x)

a(x) = q1(x)b(x) + r1(x)

v1(x) = v-1(x) – q1(x)v0(x) = 1 w1(x) = w-1(x) – q1(x)w0(x) = -q1(x)

r1(x) = a(x)v1(x) + b(x)w1(x)

r2(x) = b(x) mod r1(x) q2(x) = quotient of b(x)/r1(x)

b(x) = q2(x)r1(x) + r2(x)

v2(x) = v0(x) – q2(x)v1(x) w2(x) = w0(x) – q2(x)w1(x)

r2(x) = a(x)v2(x) + b(x)w2(x)

r3(x) = r1(x) mod r2(x) q3(x) = quotient of r1(x)/r2(x)

r1(x) = q3(x)r2(x) + r3(x)

v3(x) = v1(x) – q3(x)v2(x) w3(x) = w1(x) – q3(x)w2(x)

r3(x) = a(x)v3(x) + b(x)w3(x)

f

rn(x) = rn – 2(x) mod rn – 1(x) qn(x) = quotient of rn – 2(x)/rn – 2(x)

rn – 2(x) = qn(x)rn – 1(x) + rn(x)

vn(x) = vn – 2(x) – qn(x)vn – 1(x) wn(x) = wn – 2(x) – qn(x)wn – 1(x)

rn(x) = a(x)vn(x) + b(x)wn(x)

rn + 1(x) = rn – 1(x) mod rn(x) = 0 qn + 1(x) = quotient of rn – 1(x)/rn(x)

rn – 1(x) = qn + 1(x)rn(x) + 0

d(x) = gcd(a(x), b(x)) = rn(x) v(x) = vn(x); w(x) = wn(x)

Table 5.4 shows the calculation of the multiplicative inverse of (x7 + x + 1) mod (x8 + x4 + x3 + x + 1). The result is that (x7 + x + 1)-1 = (x7). That is, (x7 + x + 1)(x7) K 1(mod (x8 + x4 + x3 + x + 1)).

Computational Considerations

A polynomial f(x) in GF(2n)

f(x) = an – 1xn – 1 + an – 2xn – 2 + g + a1x + a0 = a n – 1

i=0 aix

i

can be uniquely represented by the sequence of its n binary coefficients (an – 1, an – 2, c , a0). Thus, every polynomial in GF(2n) can be represented by an n-bit number.

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164 CHAPTER 5 / FINITE FIELDS

ADDITION We have seen that addition of polynomials is performed by adding cor- responding coefficients, and, in the case of polynomials over Z2, addition is just the XOR operation. So, addition of two polynomials in GF(2n) corresponds to a bitwise XOR operation.

Initialization a(x) = x8 + x4 + x3 + x + 1; v-1(x) = 1; w-1(x) = 0 b(x) = x7 + x + 1; v0(x) = 0; w0(x) = 1

Iteration 1 q1(x) = x; r1(x) = x4 + x3 + x2 + 1 v1(x) = 1; w1(x) = x

Iteration 2 q2(x) = x3 + x2 + 1; r2(x) = x v2(x) = x3 + x2 + 1; w2(x) = x4 + x3 + x + 1

Iteration 3 q3(x) = x3 + x2 + x; r3(x) = 1 v3(x) = x6 + x2 + x + 1; w3(x) = x7

Iteration 4 q4(x) = x; r4(x) = 0 v4(x) = x7 + x + 1; w4(x) = x8 + x4 + x3 + x + 1

Result d(x) = r3(x) = gcd(a(x), b(x)) = 1 w(x) = w3(x) = (x7 + x + 1)-1 mod (x8 + x4 + x3 + x + 1) = x7

Table 5.4 Extended Euclid [(x8 + x4 + x3 + x + 1), (x7 + x + 1)]

Tables 5.2 and 5.3 show the addition and multiplication tables for GF(23) modulo m(x) = (x3 + x + 1). Table 5.2 uses the binary representation, and Table 5.3 uses the polynomial representation.

Consider the two polynomials in GF(28) from our earlier example:

f(x) = x6 + x4 + x2 + x + 1 and g(x) = x7 + x + 1.

(x6 + x4 + x2 + x + 1) + (x7 + x + 1) = x7 + x6 + x4 + x2 (polynomial notation) (01010111)⊕ (10000011) = (11010100) (binary notation) {57}⊕ {83} = {D4} (hexadecimal notation)7

7A basic refresher on number systems (decimal, binary, hexadecimal) can be found at the Computer Science Student Resource Site at WilliamStallings.com/StudentSupport.html. Here each of two groups of 4 bits in a byte is denoted by a single hexadecimal character, and the two characters are enclosed in brackets.

MULTIPLICATION There is no simple XOR operation that will accomplish multi- plication in GF(2n). However, a reasonably straightforward, easily implemented technique is available. We will discuss the technique with reference to GF(28) using m(x) = x8 + x4 + x3 + x + 1, which is the finite field used in AES. The technique readily generalizes to GF(2n).

The technique is based on the observation that

x8 mod m(x) = [m(x) – x8] = (x4 + x3 + x + 1) (5.4)

5.6 / FINITE FIELDS OF THE FORM GF(2n) 165

A moment’s thought should convince you that Equation (5.4) is true; if you are not sure, divide it out. In general, in GF(2n) with an nth-degree polynomial p(x), we have xn mod p(x) = [p(x) – xn].

Now, consider a polynomial in GF(28), which has the form f(x) = b7x7 + b6x6 + b5x5 + b4x4 + b3x3 + b2x2 + b1x + b0. If we multiply by x, we have

x * f(x) = (b7x8 + b6x7 + b5x6 + b4x5 + b3x4

+ b2x3 + b1x2 + b0x) mod m(x) (5.5)

If b7 = 0, then the result is a polynomial of degree less than 8, which is already in reduced form, and no further computation is necessary. If b7 = 1, then reduction modulo m(x) is achieved using Equation (5.4):

x * f(x) = (b6x7 + b5x6 + b4x5 + b3x4 + b2x3 + b1x2 + b0x) + (x4 + x3 + x + 1)

It follows that multiplication by x (i.e., 00000010) can be implemented as a 1-bit left shift followed by a conditional bitwise XOR with (00011011), which represents (x4 + x3 + x + 1). To summarize,

x * f(x) = b (b6b5b4b3b2b1b00) if b7 = 0 (b6b5b4b3b2b1b00)⊕ (00011011) if b7 = 1

(5.6)

Multiplication by a higher power of x can be achieved by repeated application of Equation (5.6). By adding intermediate results, multiplication by any constant in GF(28) can be achieved.

In an earlier example, we showed that for f(x) = x6 + x4 + x2 + x + 1, g(x) = x7 + x + 1, and m(x) = x8 + x4 + x3 + x + 1, we have f(x) * g(x) mod m(x) = x7 + x6 + 1. Redoing this in binary arithmetic, we need to compute (01010111) * (10000011). First, we determine the results of multiplication by powers of x:

(01010111) * (00000010) = (10101110) (01010111) * (00000100) = (01011100)⊕ (00011011) = (01000111) (01010111) * (00001000) = (10001110) (01010111) * (00010000) = (00011100)⊕ (00011011) = (00000111) (01010111) * (00100000) = (00001110) (01010111) * (01000000) = (00011100) (01010111) * (10000000) = (00111000)

So,

(01010111) * (10000011) = (01010111) * [(00000001)⊕ (00000010)⊕ (10000000)]

= (01010111)⊕ (10101110)⊕ (00111000) = (11000001)

which is equivalent to x7 + x6 + 1.

166 CHAPTER 5 / FINITE FIELDS

Using a Generator

An equivalent technique for defining a finite field of the form GF(2n), using the same irreducible polynomial, is sometimes more convenient. To begin, we need two definitions: A generator g of a finite field F of order q (contains q elements) is an element whose first q – 1 powers generate all the nonzero elements of F. That is, the elements of F consist of 0, g0, g1, c , gq – 2. Consider a field F defined by a polynomial f(x). An element b contained in F is called a root of the polynomial if f(b) = 0. Finally, it can be shown that a root g of an irreducible polynomial is a gen- erator of the finite field defined on that polynomial.

Power Representation

Polynomial Representation

Binary Representation

Decimal (Hex) Representation

0 0 000 0

g0(= g7) 1 001 1

g1 g 010 2

g2 g2 100 4

g3 g + 1 011 3

g4 g2 + g 110 6

g5 g2 + g + 1 111 7

g6 g2 + 1 101 5

Table 5.5 Generator for GF(23) using x3 + x + 1

Let us consider the finite field GF(23), defined over the irreducible poly- nomial x3 + x + 1, discussed previously. Thus, the generator g must satisfy f(g) = g3 + g + 1 = 0. Keep in mind, as discussed previously, that we need not find a numerical solution to this equality. Rather, we deal with polynomial arith- metic in which arithmetic on the coefficients is performed modulo 2. Therefore, the solution to the preceding equality is g3 = -g – 1 = g + 1. We now show that g in fact generates all of the polynomials of degree less than 3. We have the following.

g4 = g(g3) = g(g + 1) = g2 + g g5 = g(g4) = g(g2 + g) = g3 + g2 = g2 + g + 1 g6 = g(g5) = g(g2 + g + 1) = g3 + g2 + g = g2 + g + g + 1 = g2 + 1 g7 = g(g6) = g(g2 + 1) = g3 + g = g + g + 1 = 1 = g0

We see that the powers of g generate all the nonzero polynomials in GF(23). Also, it should be clear that gk = gk mod7 for any integer k. Table 5.5 shows the power representation, as well as the polynomial and binary representations.

5.6 / FINITE FIELDS OF THE FORM GF(2n) 167

In general, for GF(2n) with irreducible polynomial f(x), determine gn = f(g) – gn. Then calculate all of the powers of g from gn + 1 through g2

n – 2. The elements of the field correspond to the powers of g from g0 through g2

n – 2 plus the value 0. For multiplication of two elements in the field, use the equality gk = gk mod(2

n – 1) for any integer k.

Summary

In this section, we have shown how to construct a finite field of order 2n. Specifically, we defined GF(2n) with the following properties.

1. GF(2n) consists of 2n elements.

2. The binary operations + and * are defined over the set. The operations of addition, subtraction, multiplication, and division can be performed with- out leaving the set. Each element of the set other than 0 has a multiplicative inverse.

We have shown that the elements of GF(2n) can be defined as the set of all polynomials of degree n – 1 or less with binary coefficients. Each such polynomial can be represented by a unique n-bit value. Arithmetic is defined as polynomial arithmetic modulo some irreducible polynomial of degree n. We have also seen that an equivalent definition of a finite field GF(2n) makes use of a generator and that arithmetic is defined using powers of the generator.

This power representation makes multiplication easy. To multiply in the power notation, add exponents modulo 7. For example, g4 * g6 = g(10 mod 7) = g3 = g + 1. The same result is achieved using polynomial arithmetic: We have g4 = g2 + g and g6 = g2 + 1. Then, (g2 + g) * (g2 + 1) = g4 + g3 + g2 + g. Next, we need to determine (g4 + g3 + g2 + 1) mod (g3 + g + 1) by division:

g + 1 g3 + g + 1�g4 + g3 + g2 + g

g4 + g2 + g g3

g3 + g + 1 g + 1

We get a result of g + 1, which agrees with the result obtained using the power representation.

Table 5.6 shows the addition and multiplication tables for GF(23) using the power representation. Note that this yields the identical results to the polynomial representation (Table 5.3) with some of the rows and columns i nterchanged.

168 CHAPTER 5 / FINITE FIELDS

00 0

00 1

01 0

10 0

01 1

11 0

11 1

10 1

+ 0

1 G

g2 g3

g4 g5

g6

00 0

0 0

1 G

g2 g

+ 1

g2 +

g g2

+ g

+ 1

g2 +

1

00 1

1 1

0 g

+ 1

g2 +

1 g

g2 +

g +

1 g2

+ g

g2

01 0

g g

g +

1 0

g2 +

g 1

g2 g2

+ 1

g2 +

g +

1

10 0

g2 g2

g2 +

1 g2

+ g

0 g2

+ g

+ 1

g g

+ 1

1

01 1

g3 g

+ 1

g 1

g2 +

g +

1 0

g2 +

1 g2

g2 +

g

11 0

g4 g2

+ g

g2 +

g +

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+ 1

0 1

g +

1

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g5 g2

+ g

+ 1

g2 +

g g2

+ 1

g +

1 g2

1 0

g

10 1

g6 g2

+ 1

g2 g2

+ g

+ 1

1 g2

+ g

g +

1 g

0

(a )

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5.7 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 169

5.7 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS

Key Terms

abelian group associative coefficient set commutative commutative ring cyclic group divisor Euclidean algorithm field finite field finite group generator

greatest common divisor group identity element infinite field infinite group integral domain inverse element irreducible polynomial modular arithmetic modular polynomial

arithmetic monic polynomial

order polynomial polynomial arithmetic polynomial ring prime number prime polynomial relatively prime residue ring

Review Questions

5.1 Briefly define a group. 5.2 Briefly define a ring. 5.3 Briefly define a field. 5.4 List three classes of polynomial arithmetic.

Problems

5.1 For the group Sn of all permutations of n distinct symbols, a. what is the number of elements in Sn? b. show that Sn is not abelian for n 7 2.

5.2 Does the set of residue classes (mod3) form a group a. with respect to modular addition? b. with respect to modular multiplication?

5.3 Let S = {0, a, b, c}. The addition and multiplication on the set S is defined in the following tables:

+ 0 a B C 0 0 a B C

A a 0 c B

B b c 0 A

C c b a 0

* 0 a b c 0 0 0 0 0

a 0 a b c

b 0 a b c

c 0 0 0 0

Is S a noncommutative ring? Justify your answer. 5.4 Develop a set of tables similar to Table 5.1 for GF(5). 5.5 Demonstrate that the set of polynomials whose coefficients form a field is a ring. 5.6 Demonstrate whether each of these statements is true or false for polynomials over a

field.

170 CHAPTER 5 / FINITE FIELDS

a. The product of monic polynomials is monic. b. The product of polynomials of degrees m and n has degree m + n. c. The sum of polynomials of degrees m and n has degree max [m, n].

5.7 For polynomial arithmetic with coefficients in Z1 1, perform the following calculations. a. (x 2 + 2 x + 9 )(x 3 + 1 1 x 2 + x + 7 ) b. (8 x 2 + 3 x + 2 )(5 x 2 + 6 )

5.8 Determine which of the following polynomials are reducible over GF(2). a. x 2 + 1 b. x 2 + x + 1 c. x 4 + x + 1

5.9 Determine the gcd of the following pairs of polynomials. a. (x3 + 1) and (x2 + x + 1) over GF(2) b. (x3 + x + 1) and (x2 + 1) over GF(3) c. (x3 – 2x + 1) and (x2 – x – 2) over GF(5) d. (x4 + 8×3 + 7x + 8) and (2×3 + 9×2 + 10x + 1) over GF(11)

5.10 Develop a set of tables similar to Table 5.3 for GF(3) with m(x) = x2 + x + 1. 5.11 Determine the multiplicative inverse of x 2 + 1 in GF(23) with m(x) = x 3 + x – 1 . 5.12 Develop a table similar to Table 5.5 for GF(25) with m(x) = x 5 + x 4 + x 3 + x + 1 .

Programming Problems

5.13 Write a simple four-function calculator in GF(24). You may use table lookups for the multiplicative inverses.

5.14 Write a simple four-function calculator in GF(28). You should compute the multiplica- tive inverses on the fly.

171

6.1 Finite Field Arithmetic

6.2 AES Structure

General Structure Detailed Structure

6.3 AES Transformation Functions

Substitute Bytes Transformation ShiftRows Transformation MixColumns Transformation AddRoundKey Transformation

6.4 AES Key Expansion

Key Expansion Algorithm Rationale

6.5 An AES Example

Results Avalanche Effect

6.6 AES Implementation

Equivalent Inverse Cipher Implementation Aspects

6.7 Key Terms, Review Questions, and Problems

Appendix 6A Polynomials with Coefficients in GF(28)

CHAPTER

Advanced Encryption Standard

172 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD

The Advanced Encryption Standard (AES) was published by the National Institute of Standards and Technology (NIST) in 2001. AES is a symmetric block cipher that is intended to replace DES as the approved standard for a wide range of applications. Compared to public-key ciphers such as RSA, the structure of AES and most symmet- ric ciphers is quite complex and cannot be explained as easily as many other cryptographic algorithms. Accordingly, the reader may wish to begin with a simplified version of AES, which is described in Appendix I. This version allows the reader to perform encryption and decryption by hand and gain a good understanding of the working of the algorithm details. Classroom experience indicates that a study of this simplified version enhances understanding of AES.1 One possible approach is to read the chapter first, then carefully read Appendix I, and then re-read the main body of the chapter.

Appendix H looks at the evaluation criteria used by NIST to select from among the candidates for AES, plus the rationale for picking Rijndael, which was the winning candidate. This material is useful in understanding not just the AES design but also the criteria by which to judge any symmetric encryption algorithm.

6.1 FINITE FIELD ARITHMETIC

In AES, all operations are performed on 8-bit bytes. In particular, the arithmetic operations of addition, multiplication, and division are performed over the finite field GF(28). Section 5.6 discusses such operations in some detail. For the reader who has not studied Chapter 5, and as a quick review for those who have, this sec- tion summarizes the important concepts.

In essence, a field is a set in which we can do addition, subtraction, multiplica- tion, and division without leaving the set. Division is defined with the following rule: a/b = a(b-1). An example of a finite field (one with a finite number of elements) is the set Zp consisting of all the integers {0, 1, c , p – 1}, where p is a prime num- ber and in which arithmetic is carried out modulo p.

1However, you may safely skip Appendix I, at least on a first reading. If you get lost or bogged down in the details of AES, then you can go back and start with simplified AES.

LEARNING OBJECTIVES

After studying this chapter, you should be able to:

◆ Present an overview of the general structure of Advanced Encryption Standard (AES).

◆ Understand the four transformations used in AES.

◆ Explain the AES key expansion algorithm.

◆ Understand the use of polynomials with coefficients in GF(28).

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6.1 / FINITE FIELD ARITHMETIC 173

Virtually all encryption algorithms, both conventional and public-key, involve arithmetic operations on integers. If one of the operations used in the algorithm is division, then we need to work in arithmetic defined over a field; this is because division requires that each nonzero element have a multiplicative inverse. For con- venience and for implementation efficiency, we would also like to work with inte- gers that fit exactly into a given number of bits, with no wasted bit patterns. That is, we wish to work with integers in the range 0 through 2n – 1, which fit into an n-bit word. Unfortunately, the set of such integers, Z2n, using modular arithmetic, is not a field. For example, the integer 2 has no multiplicative inverse in Z2n, that is, there is no integer b, such that 2b mod 2n = 1.

There is a way of defining a finite field containing 2n elements; such a field is referred to as GF(2n). Consider the set, S, of all polynomials of degree n – 1 or less with binary coefficients. Thus, each polynomial has the form

f(x) = an – 1xn – 1 + an – 2xn – 2 + g + a1x + a0 = a n – 1

i=0 aix

i

where each ai takes on the value 0 or 1. There are a total of 2 n different polynomials

in S. For n = 3, the 23 = 8 polynomials in the set are

0 x x2 x2 + x 1 x + 1 x2 + 1 x2 + x + 1

With the appropriate definition of arithmetic operations, each such set S is a finite field. The definition consists of the following elements.

1. Arithmetic follows the ordinary rules of polynomial arithmetic using the basic rules of algebra with the following two refinements.

2. Arithmetic on the coefficients is performed modulo 2. This is the same as the XOR operation.

3. If multiplication results in a polynomial of degree greater than n – 1, then the polynomial is reduced modulo some irreducible polynomial m(x) of degree n. That is, we divide by m(x) and keep the remainder. For a polynomial f(x), the remainder is expressed as r(x) = f(x) mod m(x). A polynomial m(x) is called irreducible if and only if m(x) cannot be expressed as a product of two polynomials, both of degree lower than that of m(x).

For example, to construct the finite field GF(23), we need to choose an irre- ducible polynomial of degree 3. There are only two such polynomials: (x3 + x2 + 1) and (x3 + x + 1). Addition is equivalent to taking the XOR of like terms. Thus, (x + 1) + x = 1.

A polynomial in GF(2n) can be uniquely represented by its n binary coeffi cients (an – 1an – 2 c a0). Therefore, every polynomial in GF(2n) can be represented by an n-bit number. Addition is performed by taking the bitwise XOR of the two n-bit elements. There is no simple XOR operation that will accomplish multiplication in GF(2n). However, a reasonably straightforward, easily implemented, technique is available. In essence, it can be shown that multiplication of a number in GF(2n) by

174 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD

2 consists of a left shift followed by a conditional XOR with a constant. Multiplication by larger numbers can be achieved by repeated application of this rule.

For example, AES uses arithmetic in the finite field GF(28) with the irreducible polynomial m(x) = x8 + x4 + x3 + x + 1. Consider two elements A = (a7a6 c a1a0) and B = (b7b6 c b1b0). The sum A + B = (c7c6 c c1c0), where ci = ai⊕ bi. The multiplication {02} # A equals (a6 c a1a00) if a7 = 0 and equals (a6 c a1a00)⊕ (00011011) if a7 = 1.2

To summarize, AES operates on 8-bit bytes. Addition of two bytes is defined as the bitwise XOR operation. Multiplication of two bytes is defined as multiplica- tion in the finite field GF(28), with the irreducible polynomial3 m(x) = x8 + x4 + x3 + x + 1. The developers of Rijndael give as their motivation for selecting this one of the 30 possible irreducible polynomials of degree 8 that it is the first one on the list given in [LIDL94].

6.2 AES STRUCTURE

General Structure

Figure 6.1 shows the overall structure of the AES encryption process. The cipher takes a plaintext block size of 128 bits, or 16 bytes. The key length can be 16, 24, or 32 bytes (128, 192, or 256 bits). The algorithm is referred to as AES-128, AES-192, or AES-256, depending on the key length.

The input to the encryption and decryption algorithms is a single 128-bit block. In FIPS PUB 197, this block is depicted as a 4 * 4 square matrix of bytes. This block is copied into the State array, which is modified at each stage of encryption or decryption. After the final stage, State is copied to an output matrix. These opera- tions are depicted in Figure 6.2a. Similarly, the key is depicted as a square matrix of bytes. This key is then expanded into an array of key schedule words. Figure 6.2b shows the expansion for the 128-bit key. Each word is four bytes, and the total key schedule is 44 words for the 128-bit key. Note that the ordering of bytes within a ma- trix is by column. So, for example, the first four bytes of a 128-bit plaintext input to the encryption cipher occupy the first column of the in matrix, the second four bytes occupy the second column, and so on. Similarly, the first four bytes of the expanded key, which form a word, occupy the first column of the w matrix.

The cipher consists of N rounds, where the number of rounds depends on the key length: 10 rounds for a 16-byte key, 12 rounds for a 24-byte key, and 14 rounds for a 32-byte key (Table 6.1). The first N – 1 rounds consist of four distinct trans- formation functions: SubBytes, ShiftRows, MixColumns, and AddRoundKey, which are described subsequently. The final round contains only three transforma- tions, and there is a initial single transformation (AddRoundKey) before the first round, which can be considered Round 0. Each transformation takes one or more

2In FIPS PUB 197, a hexadecimal number is indicated by enclosing it in curly brackets. We use that convention in this chapter. 3In the remainder of this discussion, references to GF(28) refer to the finite field defined with this polynomial.

6.2 / AES STRUCTURE 175

Figure 6.1 AES Encryption Process

Initial transformation

K ey

e xp

an si

on

Plaintext—16 bytes (128 bits) Key—M bytes

Key (M bytes)Round 0 key

(16 bytes)

Round 1 key (16 bytes)

Round N – 1 key (16 bytes)

Round N key (16 bytes)

Cipehertext—16 bytes (128 bits)

No. of rounds

10 16

Key Length (bytes)

Input state (16 bytes)

State after initial

transformation (16 bytes)

Final state (16 bytes)

Round N – 1 output state (16 bytes)

Round 1 output state (16 bytes)

Round 1 (4 transformations)

Round N – 1 (4 transformations)

Round N (3 transformations)

12 24

14 32

4 * 4 matrices as input and produces a 4 * 4 matrix as output. Figure 6.1 shows that the output of each round is a 4 * 4 matrix, with the output of the final round being the ciphertext. Also, the key expansion function generates N + 1 round keys, each of which is a distinct 4 * 4 matrix. Each round key serves as one of the inputs to the AddRoundKey transformation in each round.

F ig

ur e

6. 2

A E

S D

at a

St ru

ct ur

es

in 0

in 4

in 8

in 12

in 1

in 5

in 9

in 13

in 2

in 6

in 10

in 14

in 3

in 7

in 11

in 15

k 0

w 0

w 1

w 2

w 43

w 42

k 4 k 8

k 1 2

k 1 k 5

k 9 k 1

3

k 2 k 6

k 1 0

k 1 4

k 3 k 7

k 1 1

k 1 5

ou t 0

ou t 4

ou t 8

ou t 1

2

ou t 1

ou t 5

ou t 9

ou t 1

3

ou t 2

ou t 6

ou t 1

0 ou

t 1 4

ou t 3

ou t 7

ou t 1

1 ou

t 1 5

s 0 ,0

s 1 ,0

s 2 ,0

s 3 ,0

s 0 ,1

s 1 ,1

s 2 ,1

s 3 ,1

s 0 ,2

s 1 ,2

s 2 ,2

s 3 ,2

s 0 ,3

s 1 ,3

s 2 ,3

s 3 ,3

s 0 ,0

s 1 ,0

s 2 ,0

s 3 ,0

s 0 ,1

s 1 ,1

s 2 ,1

s 3 ,1

s 0 ,2

s 1 ,2

s 2 ,2

s 3 ,2

s 0 ,3

s 1 ,3

s 2 ,3

s 3 ,3

(a ) I

np ut

, s ta

te a

rr ay

, a nd

o ut

pu t

(b ) K

ey a

nd e

xp an

de d

ke y

176

6.2 / AES STRUCTURE 177

Key Size (words/bytes/bits) 4/16/128 6/24/192 8/32/256 Plaintext Block Size (words/bytes/bits) 4/16/128 4/16/128 4/16/128 Number of Rounds 10 12 14 Round Key Size (words/bytes/bits) 4/16/128 4/16/128 4/16/128 Expanded Key Size (words/bytes) 44/176 52/208 60/240

Table 6.1 AES Parameters

Detailed Structure

Figure 6.3 shows the AES cipher in more detail, indicating the sequence of transfor- mations in each round and showing the corresponding decryption function. As was done in Chapter 4, we show encryption proceeding down the page and decryption proceeding up the page.

Before delving into details, we can make several comments about the overall AES structure.

1. One noteworthy feature of this structure is that it is not a Feistel structure. Recall that, in the classic Feistel structure, half of the data block is used to modify the other half of the data block and then the halves are swapped. AES instead processes the entire data block as a single matrix during each round using substitutions and permutation.

2. The key that is provided as input is expanded into an array of forty-four 32-bit words, w[i]. Four distinct words (128 bits) serve as a round key for each round; these are indicated in Figure 6.3.

3. Four different stages are used, one of permutation and three of substitution:

■ Substitute bytes: Uses an S-box to perform a byte-by-byte substitution of the block.

■ ShiftRows: A simple permutation.

■ MixColumns: A substitution that makes use of arithmetic over GF(28).

■ AddRoundKey: A simple bitwise XOR of the current block with a portion of the expanded key.

4. The structure is quite simple. For both encryption and decryption, the cipher begins with an AddRoundKey stage, followed by nine rounds that each in- cludes all four stages, followed by a tenth round of three stages. Figure 6.4 depicts the structure of a full encryption round.

5. Only the AddRoundKey stage makes use of the key. For this reason, the cipher begins and ends with an AddRoundKey stage. Any other stage, applied at the beginning or end, is reversible without knowledge of the key and so would add no security.

6. The AddRoundKey stage is, in effect, a form of Vernam cipher and by itself would not be formidable. The other three stages together provide confusion, diffusion, and nonlinearity, but by themselves would provide no security be- cause they do not use the key. We can view the cipher as alternating operations of XOR encryption (AddRoundKey) of a block, followed by scrambling of the

178 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD

Figure 6.3 AES Encryption and Decryption

Add round key

w[4, 7]

Plaintext (16 bytes)

Plaintext (16 bytes)

Substitute bytes

Expand key

Shift rows

Mix columnsR ou

nd 1

R ou

nd 9

R ou

nd 1

0

Add round key

Substitute bytes

Shift rows

Mix columns

Add round key

Substitute bytes

Shift rows

Add round key

Ciphertext (16 bytes)

(a) Encryption

Key (16 bytes)

Add round key

Inverse sub bytes

Inverse shift rows

Inverse mix cols

R ou

nd 1

0 R

ou nd

9 R

ou nd

1

Add round key

Inverse sub bytes

Inverse shift rows

Inverse mix cols

Add round key

Inverse sub bytes

Inverse shift rows

Add round key

Ciphertext (16 bytes)

(b) Decryption

w[36, 39]

w[40, 43]

w[0, 3]

block (the other three stages), followed by XOR encryption, and so on. This scheme is both efficient and highly secure.

7. Each stage is easily reversible. For the Substitute Byte, ShiftRows, and MixColumns stages, an inverse function is used in the decryption algorithm. For the AddRoundKey stage, the inverse is achieved by XORing the same round key to the block, using the result that A⊕ B⊕ B = A.

8. As with most block ciphers, the decryption algorithm makes use of the expanded key in reverse order. However, the decryption algorithm is not

6.3 / AES TRANSFORMATION FUNCTIONS 179

Figure 6.4 AES Encryption Round

SSubBytes

State

State

State

State

State

ShiftRows

MixColumns

AddRoundKey

S S S S S S S S S S S S S S S

M M M M

r0 r1 r2 r3 r4 r5 r6 r7 r8 r9 r10 r11 r12 r13 r14 r15

identical to the encryption algorithm. This is a consequence of the particular structure of AES.

9. Once it is established that all four stages are reversible, it is easy to verify that decryption does recover the plaintext. Figure 6.3 lays out encryption and decryption going in opposite vertical directions. At each horizontal point (e.g., the dashed line in the figure), State is the same for both encryption and decryption.

10. The final round of both encryption and decryption consists of only three stages. Again, this is a consequence of the particular structure of AES and is required to make the cipher reversible.

6.3 AES TRANSFORMATION FUNCTIONS

We now turn to a discussion of each of the four transformations used in AES. For each stage, we describe the forward (encryption) algorithm, the inverse ( decryption) algorithm, and the rationale for the stage.

180 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD

Substitute Bytes Transformation

FORWARD AND INVERSE TRANSFORMATIONS The forward substitute byte transformation, called SubBytes, is a simple table lookup (Figure 6.5a). AES defines a 16 * 16 matrix of byte values, called an S-box (Table 6.2a), that con- tains a permutation of all possible 256 8-bit values. Each individual byte of State is mapped into a new byte in the following way: The leftmost 4 bits of the byte are used as a row value and the rightmost 4 bits are used as a column value. These row and column values serve as indexes into the S-box to select a unique 8-bit output value. For example, the hexadecimal value {95} references row 9, column 5 of the S-box, which contains the value {2A}. Accordingly, the value {95} is mapped into the value {2A}.

Figure 6.5 AES Byte-Level Operations

s0,0 s0,1 s0,2 s0,3

s1,0 s1,2 s1,3

s2,0 s2,1 s2,2 s2,3

s3,0 s3,1 s3,2 s3,3

s0,0 s0,1 s0,2 s0,3

s1,0 s1,2 s1,3

s2,0 s2,1 s2,2 s2,3

s3,0 s3,1 s3,2 s3,3

(b) Add round key transformation

(a) Substitute byte transformation

S-box

x

y

¿ ¿ ¿ ¿

¿ ¿¿¿

s1,1

s0,0

wi wi+2 wi+3

s0,2 s0,3

s1,0 s1,2 s1,3 =

s2,0 s2,2 s2,3

s3,0 s3,2 s3,3

s1,1

s0,0 s0,2 s0,3

s1,0 s1,2 s1,3

s2,0 s2,2 s2,3

s3,0 s3,2 s3,3

s1,1

s0,1

s2,1

s3,1

wi+1

s0,1

s2,1

s3,1

s1,1

¿¿¿

¿ ¿ ¿ ¿

¿

¿

¿

¿ ¿

¿ ¿

¿ ¿ ¿ ¿

¿ ¿ ¿

¿ ¿ ¿

6.3 / AES TRANSFORMATION FUNCTIONS 181

y 0 1 2 3 4 5 6 7 8 9 A B C D E F

0 63 7C 77 7B F2 6B 6F C5 30 01 67 2B FE D7 AB 76 1 CA 82 C9 7D FA 59 47 F0 AD D4 A2 AF 9C A4 72 C0 2 B7 FD 93 26 36 3F F7 CC 34 A5 E5 F1 71 D8 31 15 3 04 C7 23 C3 18 96 05 9A 07 12 80 E2 EB 27 B2 75 4 09 83 2C 1A 1B 6E 5A A0 52 3B D6 B3 29 E3 2F 84 5 53 D1 00 ED 20 FC B1 5B 6A CB BE 39 4A 4C 58 CF 6 D0 EF AA FB 43 4D 33 85 45 F9 02 7F 50 3C 9F A8

x 7 51 A3 40 8F 92 9D 38 F5 BC B6 DA 21 10 FF F3 D2 8 CD 0C 13 EC 5F 97 44 17 C4 A7 7E 3D 64 5D 19 73 9 60 81 4F DC 22 2A 90 88 46 EE B8 14 DE 5E 0B DB A E0 32 3A 0A 49 06 24 5C C2 D3 AC 62 91 95 E4 79 B E7 C8 37 6D 8D D5 4E A9 6C 56 F4 EA 65 7A AE 08 C BA 78 25 2E 1C A6 B4 C6 E8 DD 74 1F 4B BD 8B 8A D 70 3E B5 66 48 03 F6 0E 61 35 57 B9 86 C1 1D 9E E E1 F8 98 11 69 D9 8E 94 9B 1E 87 E9 CE 55 28 DF F 8C A1 89 0D BF E6 42 68 41 99 2D 0F B0 54 BB 16

(a) S-box

y 0 1 2 3 4 5 6 7 8 9 A B C D E F

0 52 09 6A D5 30 36 A5 38 BF 40 A3 9E 81 F3 D7 FB 1 7C E3 39 82 9B 2F FF 87 34 8E 43 44 C4 DE E9 CB 2 54 7B 94 32 A6 C2 23 3D EE 4C 95 0B 42 FA C3 4E 3 08 2E A1 66 28 D9 24 B2 76 5B A2 49 6D 8B D1 25 4 72 F8 F6 64 86 68 98 16 D4 A4 5C CC 5D 65 B6 92 5 6C 70 48 50 FD ED B9 DA 5E 15 46 57 A7 8D 9D 84 6 90 D8 AB 00 8C BC D3 0A F7 E4 58 05 B8 B3 45 06

x 7 D0 2C 1E 8F CA 3F 0F 02 C1 AF BD 03 01 13 8A 6B 8 3A 91 11 41 4F 67 DC EA 97 F2 CF CE F0 B4 E6 73 9 96 AC 74 22 E7 AD 35 85 E2 F9 37 E8 1C 75 DF 6E A 47 F1 1A 71 1D 29 C5 89 6F B7 62 0E AA 18 BE 1B B FC 56 3E 4B C6 D2 79 20 9A DB C0 FE 78 CD 5A F4 C 1F DD A8 33 88 07 C7 31 B1 12 10 59 27 80 EC 5F D 60 51 7F A9 19 B5 4A 0D 2D E5 7A 9F 93 C9 9C EF E A0 E0 3B 4D AE 2A F5 B0 C8 EB BB 3C 83 53 99 61 F 17 2B 04 7E BA 77 D6 26 E1 69 14 63 55 21 0C 7D

(b) Inverse S-box

Table 6.2 AES S-Boxes

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182 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD

Here is an example of the SubBytes transformation:

EA 04 65 85 87 F2 4D 97

83 45 5D 96 EC 6E 4C 90

5C 33 98 B0 S 4A C3 46 E7 F0 2D AD C5 8C D8 95 A6

The S-box is constructed in the following fashion (Figure 6.6a).

Figure 6.6 Constuction of S-Box and IS-Box

b0 b1 b2 b3 b4 b5 b6 b7

=

1 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1

b0 b1 b2 b3 b4 b5 b6 b7

+

1 1 0 0 0 1 1 0

Inverse in GF(28)

Byte to bit column vector

Bit column vector to byte

Byte at row y, column x

initialized to yx yx

S(yx)

(a) Calculation of byte at row y, column x of S-box

(a) Calculation of byte at row y, column x of IS-box

Inverse in GF(28)

Byte to bit column vector

Bit column vector to byte

Byte at row y, column x

initialized to yx yx

b0¿

b¿

b¿ b ¿

1

2

3

b4 b5 b6 b7

=

0 0 1 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1 1 0 0 1 0 1 0 0 0 1 0 0 1 0 1 0

b0 b1 b2 b3 b4 b5 b6 b7

+

1 0 1 0 0 0 0 0

IS(yx)

¿

¿ ¿

¿

¿

¿

¿

¿

¿

¿

¿ ¿

6.3 / AES TRANSFORMATION FUNCTIONS 183

1. Initialize the S-box with the byte values in ascending sequence row by row. The first row contains {00}, {01}, {02}, c , {0F}; the second row contains {10}, {11}, etc.; and so on. Thus, the value of the byte at row y, column x is {yx}.

2. Map each byte in the S-box to its multiplicative inverse in the finite field GF(28); the value {00} is mapped to itself.

3. Consider that each byte in the S-box consists of 8 bits labeled (b7, b6, b5, b4, b3, b2, b1, b0). Apply the following transformation to each bit of each byte in the S-box:

bi = = bi⊕ b(i+ 4) mod 8 ⊕ b(i+ 5) mod 8 ⊕ b(i+ 6) mod 8 ⊕ b(i+ 7) mod 8 ⊕ ci (6.1)

where ci is the ith bit of byte c with the value {63}; that is, (c7c6c5c4c3c2c1c0) = (01100011). The prime (′) indicates that the variable is to be updated by the value on the right. The AES standard depicts this transfor- mation in matrix form as follows.

H b0 =

b1 =

b2 =

b3 =

b4 =

b5 =

b6 =

b7 =

X = H 1 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1

X H b0 b1 b2 b3 b4 b5 b6 b7

X + H 1 1 0 0 0 1 1 0

X (6.2) Equation (6.2) has to be interpreted carefully. In ordinary matrix multiplica-

tion,4 each element in the product matrix is the sum of products of the elements of one row and one column. In this case, each element in the product matrix is the bitwise XOR of products of elements of one row and one column. Furthermore, the final addition shown in Equation (6.2) is a bitwise XOR. Recall from Section 5.6 that the bitwise XOR is addition in GF(28).

As an example, consider the input value {95}. The multiplicative inverse in GF(28) is {95}-1 = {8A}, which is 10001010 in binary. Using Equation (6.2),

H 1 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1

X H 0 1 0 1 0 0 0 1

X ⊕ H 1 1 0 0 0 1 1 0

X = H 1 0 0 1 0 0 1 0

X ⊕ H 1 1 0 0 0 1 1 0

X = H 0 1 0 1 0 1 0 0

X

4For a brief review of the rules of matrix and vector multiplication, refer to Appendix E.

184 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD

The result is {2A}, which should appear in row {09} column {05} of the S-box. This is verified by checking Table 6.2a.

The inverse substitute byte transformation, called InvSubBytes, makes use of the inverse S-box shown in Table 6.2b. Note, for example, that the input {2A} produces the output {95}, and the input {95} to the S-box produces {2A}. The inverse S-box is constructed (Figure 6.6b) by applying the inverse of the transformation in Equation (6.1) followed by taking the multiplicative inverse in GF(28). The inverse transformation is

bi = = b(i+ 2) mod 8 ⊕ b(i+ 5) mod 8 ⊕ b(i+ 7) mod 8 ⊕ di

where byte d = {05}, or 00000101. We can depict this transformation as follows.

H b0 =

b1 =

b2 =

b3 =

b4 =

b5 =

b6 =

b7 =

X = H 0 0 1 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1 1 0 0 1 0 1 0 0 0 1 0 0 1 0 1 0

X H b0 b1 b2 b3 b4 b5 b6 b7

X + H 1 0 1 0 0 0 0 0

X To see that InvSubBytes is the inverse of SubBytes, label the matrices in

SubBytes and InvSubBytes as X and Y, respectively, and the vector versions of con- stants c and d as C and D, respectively. For some 8-bit vector B, Equation (6.2) becomes B= = XB⊕ C. We need to show that Y(XB⊕ C)⊕D = B. To multiply out, we must show YXB⊕ YC⊕D = B. This becomes

H 0 0 1 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1 1 0 0 1 0 1 0 0 0 1 0 0 1 0 1 0

X H 1 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1

X H b0 b1 b2 b3 b4 b5 b6 b7

X ⊕

H 0 0 1 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1 1 0 0 1 0 1 0 0 0 1 0 0 1 0 1 0

X H 1 1 0 0 0 1 1 0

X ⊕ H 1 0 1 0 0 0 0 0

X =

6.3 / AES TRANSFORMATION FUNCTIONS 185

H 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1

X H b0 b1 b2 b3 b4 b5 b6 b7

X ⊕ H 1 0 1 0 0 0 0 0

X ⊕ H 1 0 1 0 0 0 0 0

X = H b0 b1 b2 b3 b4 b5 b6 b7

X We have demonstrated that YX equals the identity matrix, and the YC = D,

so that YC⊕D equals the null vector.

RATIONALE The S-box is designed to be resistant to known cryptanalytic attacks. Specifically, the Rijndael developers sought a design that has a low correlation between input bits and output bits and the property that the output is not a linear mathematical function of the input [DAEM01]. The nonlinearity is due to the use of the multiplicative inverse. In addition, the constant in Equation (6.1) was chosen so that the S-box has no fixed points [S@box(a) = a] and no “opposite fixed points” [S@box(a) = a], where a is the bitwise complement of a.

Of course, the S-box must be invertible, that is, IS@box[S@box(a)] = a. However, the S-box does not self-inverse in the sense that it is not true that S@box(a) = IS@box(a). For example, S@box({95}) = {2A}, but IS@box({95}) = {AD}.

ShiftRows Transformation

FORWARD AND INVERSE TRANSFORMATIONS The forward shift row transformation, called ShiftRows, is depicted in Figure 6.7a. The first row of State is not altered. For the second row, a 1-byte circular left shift is performed. For the third row, a 2-byte circular left shift is performed. For the fourth row, a 3-byte circular left shift is per- formed. The following is an example of ShiftRows.

87 F2 4D 97 87 F2 4D 97

EC 6E 4C 90 6E 4C 90 EC

4A C3 46 E7 S 46 E7 4A C3 8C D8 95 A6 A6 8C D8 95

The inverse shift row transformation, called InvShiftRows, performs the cir- cular shifts in the opposite direction for each of the last three rows, with a 1-byte circular right shift for the second row, and so on.

RATIONALE The shift row transformation is more substantial than it may first appear. This is because the State, as well as the cipher input and output, is treated as an array of four 4-byte columns. Thus, on encryption, the first 4 bytes of the plaintext are copied to the first column of State, and so on. Furthermore, as will be seen, the round key is applied to State column by column. Thus, a row shift moves an individual byte from one column to another, which is a linear

186 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD

5We follow the convention of FIPS PUB 197 and use the symbol # to indicate multiplication over the finite field GF(28) and ⊕ to indicate bitwise XOR, which corresponds to addition in GF(28).

distance of a multiple of 4 bytes. Also note that the transformation ensures that the 4 bytes of one column are spread out to four different columns. Figure 6.4 illustrates the effect.

MixColumns Transformation

FORWARD AND INVERSE TRANSFORMATIONS The forward mix column transformation, called MixColumns, operates on each column individually. Each byte of a column is mapped into a new value that is a function of all four bytes in that column. The transformation can be defined by the following matrix multiplication on State (Figure 6.7b):

D02 03 01 0101 02 03 01 01 01 02 03 03 01 01 02

T D s0,0 s0,1 s0,2 s0,3s1,0 s1,1 s1,2 s1,3 s2,0 s2,1 s2,2 s2,3 s3,0 s3,1 s3,2 s3,3

T = D s0,0= s0,1= s0,2= s0,3=s1,0= s1,1= s1,2= s1,3= s2,0 = s2,1

= s2,2 = s2,3

=

s3,0 = s3,1

= s3,2 = s3,3

=

T (6.3) Each element in the product matrix is the sum of products of elements of one row and one column. In this case, the individual additions and multiplications5 are

Figure 6.7 AES Row and Column Operations

s0,0 s0,1 s0,2 s0,3

s1,0 s1,1 s1,2 s1,3

s2,0 s2,1 s2,2 s2,3

s3,0 s3,1 s3,2 s3,3

s0,0 s0,1 s0,2 s0,3

s1,0 s1,1 s1,2 s1,3

s2,0 s2,1 s2,2 s2,3

s3,0 s3,1 s3,2 s3,3

s0,0 s0,1 s0,2 s0,3

s1,0 s1,1 s1,2 s1,3

s2,0 s2,1 s2,2 s2,3

s3,0 s3,1 s3,2 s3,3

s0,0 s0,1 s0,2 s0,3

s1,1 s1,2 s1,3 s1,0

s2,2 s2,3 s2,0 s2,1

s3,3 s3,0 s3,1 s3,2

(a) Shift row transformation

(b) Mix column transformation

2 3 1 1 1 2 3 1 1 1 2 3 3 1 1 2

=*

¿ ¿ ¿ ¿

¿¿¿¿

¿ ¿ ¿ ¿

¿¿¿¿

6.3 / AES TRANSFORMATION FUNCTIONS 187

performed in GF(28). The MixColumns transformation on a single column of State can be expressed as

s0, j = = (2 # s0, j)⊕ (3 # s1, j)⊕ s2, j⊕ s3, j

s1, j = = s0, j⊕ (2 # s1, j)⊕ (3 # s2, j)⊕ s3, j

s2, j = = s0, j⊕ s1, j⊕ (2 # s2, j)⊕ (3 # s3, j)

s3, j = = (3 # s0, j)⊕ s1, j⊕ s2, j⊕ (2 # s3, j)

(6.4)

The following is an example of MixColumns:

87 F2 4D 97 47 40 A3 4C

6E 4C 90 EC 37 D4 70 9F

46 E7 4A C3 S 94 E4 3A 42 A6 8C D8 95 ED A5 A6 BC

Let us verify the first column of this example. Recall from Section 5.6 that, in GF(28), addition is the bitwise XOR operation and that multiplication can be per- formed according to the rule established in Equation (4.14). In particular, multipli- cation of a value by x (i.e., by {02}) can be implemented as a 1-bit left shift followed by a conditional bitwise XOR with (0001 1011) if the leftmost bit of the original value (prior to the shift) is 1. Thus, to verify the MixColumns transformation on the first column, we need to show that

({02} # {87}) ⊕ ({03} # {6E}) ⊕ {46} ⊕ {A6} = {47} {87} ⊕ ({02} # {6E}) ⊕ ({03} # {46}) ⊕ {A6} = {37} {87} ⊕ {6E} ⊕ ({02} # {46}) ⊕ ({03} # {A6}) = {94} ({03} # {87}) ⊕ {6E} ⊕ {46} ⊕ ({02} # {A6}) = {ED} For the first equation, we have {02} # {87} = (0000 1110)⊕ (0001 1011) =

(0001 0101) and {03} # {6E} = {6E}⊕ ({02} # {6E}) = (0110 1110)⊕ (1101 1100) = (1011 0010). Then,

{02} # {87} = 0001 0101 {03} # {6E} = 1011 0010 {46} = 0100 0110 {A6} = 1010 0110

0100 0111 = {47}

The other equations can be similarly verified. The inverse mix column transformation, called InvMixColumns, is defined by

the following matrix multiplication:

D 0E 0B 0D 0909 0E 0B 0D 0D 09 0E 0B 0B 0D 09 0E

T D s0,0 s0,1 s0,2 s0,3s1,0 s1,1 s1,2 s1,3 s2,0 s2,1 s2,2 s2,3 s3,0 s3,1 s3,2 s3,3

T = D s0,0= s0,1= s0,2= s0,3=s1,0= s1,1= s1,2= s1,3= s2,0 = s2,1

= s2,2 = s2,3

=

s3,0 = s3,1

= s3,2 = s3,3

=

T (6.5)

188 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD

It is not immediately clear that Equation (6.5) is the inverse of Equation (6.3). We need to show

D 0E 0B 0D 0909 0E 0B 0D 0D 09 0E 0B 0B 0D 09 0E

T D02 03 01 0101 02 03 01 01 01 02 03 03 01 01 02

T D s0,0 s0,1 s0,2 s0,3s1,0 s1,1 s1,2 s1,3 s2,0 s2,1 s2,2 s2,3 s3,0 s3,1 s3,2 s3,3

T = D s0,0 s0,1 s0,2 s0,3s1,0 s1,1 s1,2 s1,3 s2,0 s2,1 s2,2 s2,3 s0,3 s3,1 s3,2 s3,3

T which is equivalent to showing

D 0E 0B 0D 0909 0E 0B 0D 0D 09 0E 0B 0B 0D 09 0E

T D02 03 01 0101 02 03 01 01 01 02 03 03 01 01 02

T = D1 0 0 00 1 0 0 0 0 1 0 0 0 0 1

T (6.6) That is, the inverse transformation matrix times the forward transformation matrix equals the identity matrix. To verify the first column of Equation (6.6), we need to show

({0E} # {02})⊕ {0B}⊕ {0D}⊕ ({09} # {03}) = {01} ({09} # {02})⊕ {0E}⊕ {0B}⊕ ({0D} # {03}) = {00} ({0D} # {02})⊕ {09}⊕ {0E}⊕ ({0B} # {03}) = {00}

({0B} # {02})⊕ {0D}⊕ {09}⊕ ({0E} # {03}) = {00} For the first equation, we have {0E} # {02} = 00011100 and {09} # {03} =

{09}⊕ ({09} # {02}) = 00001001⊕ 00010010 = 00011011. Then

{0E} # {02} = 00011100 {0B} = 00001011 {0D} = 00001101 {09} # {03} = 00011011

00000001

The other equations can be similarly verified. The AES document describes another way of characterizing the MixColumns

transformation, which is in terms of polynomial arithmetic. In the standard, MixColumns is defined by considering each column of State to be a four-term poly- nomial with coefficients in GF(28). Each column is multiplied modulo (x4 + 1) by the fixed polynomial a(x), given by

a(x) = {03}x3 + {01}x2 + {01}x + {02} (6.7)

Appendix 5A demonstrates that multiplication of each column of State by a(x) can be written as the matrix multiplication of Equation (6.3). Similarly, it can be seen that the transformation in Equation (6.5) corresponds to treating

6.3 / AES TRANSFORMATION FUNCTIONS 189

each column as a four-term polynomial and multiplying each column by b(x), given by

b(x) = {0B}x3 + {0D}x2 + {09}x + {0E} (6.8)

It readily can be shown that b(x) = a-1(x) mod (x4 + 1).

RATIONALE The coefficients of the matrix in Equation (6.3) are based on a linear code with maximal distance between code words, which ensures a good mixing among the bytes of each column. The mix column transformation combined with the shift row transformation ensures that after a few rounds all output bits depend on all input bits. See [DAEM99] for a discussion.

In addition, the choice of coefficients in MixColumns, which are all {01}, {02}, or {03}, was influenced by implementation considerations. As was discussed, multi- plication by these coefficients involves at most a shift and an XOR. The coefficients in InvMixColumns are more formidable to implement. However, encryption was deemed more important than decryption for two reasons:

1. For the CFB and OFB cipher modes (Figures 7.5 and 7.6; described in Chapter 7), only encryption is used.

2. As with any block cipher, AES can be used to construct a message authentica- tion code (Chapter 13), and for this, only encryption is used.

AddRoundKey Transformation

FORWARD AND INVERSE TRANSFORMATIONS In the forward add round key transfor- mation, called AddRoundKey, the 128 bits of State are bitwise XORed with the 128 bits of the round key. As shown in Figure 6.5b, the operation is viewed as a columnwise operation between the 4 bytes of a State column and one word of the round key; it can also be viewed as a byte-level operation. The following is an example of AddRoundKey:

47 40 A3 4C AC 19 28 57 EB 59 8B 1B

37 D4 70 9F 77 FA D1 5C 40 2E A1 C3

94 E4 3A 42 ⊕ 66 DC 29 00 = F2 38 13 42

ED A5 A6 BC F3 21 41 6A 1E 84 E7 D6

The first matrix is State, and the second matrix is the round key. The inverse add round key transformation is identical to the forward add

round key transformation, because the XOR operation is its own inverse.

RATIONALE The add round key transformation is as simple as possible and affects every bit of State. The complexity of the round key expansion, plus the complexity of the other stages of AES, ensure security.

Figure 6.8 is another view of a single round of AES, emphasizing the mecha- nisms and inputs of each transformation.

190 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD

6.4 AES KEY EXPANSION

Key Expansion Algorithm

The AES key expansion algorithm takes as input a four-word (16-byte) key and produces a linear array of 44 words (176 bytes). This is sufficient to provide a four- word round key for the initial AddRoundKey stage and each of the 10 rounds of the cipher. The pseudocode on the next page describes the expansion.

The key is copied into the first four words of the expanded key. The remain- der of the expanded key is filled in four words at a time. Each added word w[i] depends on the immediately preceding word, w[i – 1], and the word four positions back, w[i – 4]. In three out of four cases, a simple XOR is used. For a word whose position in the w array is a multiple of 4, a more complex function is used. Figure 6.9 illustrates the generation of the expanded key, using the symbol g to represent that complex function. The function g consists of the following subfunctions.

Figure 6.8 Inputs for Single AES Round

SubBytes

State matrix at beginning

of round

State matrix at end

of round

MixColumns matrix Round

key

Variable inputConstant inputs

ShiftRows

MixColumns

AddRoundKey

S-box

02 03 01 01 01 02 03 01 01 01 02 03 03 01 01 02

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6.4 / AES KEY EXPANSION 191

KeyExpansion (byte key[16], word w[44]) { word temp for (i = 0; i 0 and Z ∈ Z2/

P = M }pad[r](|M|) s = 0b

for i = 0 to |P|r − 1 do s = s⊕ (Pi }0b − r) s = f(s) end for Z =:s;r while |Z|r r s public key, A can obtain X2>s public key from its certificate and verify it by means of X1>s signature on the certificate.

Step 2 A then goes back to the directory and obtains the certificate of B signed by X2. Because A now has a trusted copy of X2>s public key, A can verify the signature and securely obtain B’s public key.

A has used a chain of certificates to obtain B’s public key. In the notation of X.509, this chain is expressed as

X1 V X2 W X2 V B W

In the same fashion, B can obtain A’s public key with the reverse chain:

X2 V X1 W X1 V A W

This scheme need not be limited to a chain of two certificates. An arbitrarily long path of CAs can be followed to produce a chain. A chain with N elements would be expressed as

X1 V X2 W X2 V X3 W c XN V B W

In this case, each pair of CAs in the chain (Xi, Xi+ 1) must have created certifi- cates for each other.

All these certificates of CAs by CAs need to appear in the directory, and the user needs to know how they are linked to follow a path to another user’s public-key certificate. X.509 suggests that CAs be arranged in a hierarchy so that navigation is straightforward.

Figure 14.16, taken from X.509, is an example of such a hierarchy. The con- nected circles indicate the hierarchical relationship among the CAs; the associated boxes indicate certificates maintained in the directory for each CA entry. The direc- tory entry for each CA includes two types of certificates:

■ Forward certificates: Certificates of X generated by other CAs

■ Reverse certificates: Certificates generated by X that are the certificates of other CAs

In this example, user A can acquire the following certificates from the direc- tory to establish a certification path to B:

X V W W W V V W V V Y W Y V Z W Z V B W

When A has obtained these certificates, it can unwrap the certification path in sequence to recover a trusted copy of B’s public key. Using this public key, A can send encrypted messages to B. If A wishes to receive encrypted messages back

464 CHAPTER 14 / KEY MANAGEMENT AND DISTRIBUTION

from B, or to sign messages sent to B, then B will require A’s public key, which can be obtained from the following certification path:

Z V Y W Y V V W V V W W W V X W X V A W

B can obtain this set of certificates from the directory, or A can provide them as part of its initial message to B.

REVOCATION OF CERTIFICATES Recall from Figure 14.15 that each certificate includes a period of validity, much like a credit card. Typically, a new certificate is issued just before the expiration of the old one. In addition, it may be desirable on occasion to revoke a certificate before it expires, for one of the following reasons.

1. The user’s private key is assumed to be compromised.

2. The user is no longer certified by this CA. Reasons for this include that the subject’s name has changed, the certificate is superseded, or the certificate was not issued in conformance with the CA’s policies.

3. The CA’s certificate is assumed to be compromised.

Each CA must maintain a list consisting of all revoked but not expired certificates issued by that CA, including both those issued to users and to other CAs. These lists should also be posted on the directory.

Figure 14.16 X.509 Hierarchy: A Hypothetical Example

U

V

W Y

Z

B

X

C A

U> V>

V> W>

V> Y>

W> X> X>

Y> Z> Z>

X> X> Z>

14.4 / X.509 CERTIFICATES 465

Each certificate revocation list (CRL) posted to the directory is signed by the issuer and includes (Figure 14.15b) the issuer’s name, the date the list was created, the date the next CRL is scheduled to be issued, and an entry for each revoked certificate. Each entry consists of the serial number of a certificate and revocation date for that certificate. Because serial numbers are unique within a CA, the serial number is sufficient to identify the certificate.

When a user receives a certificate in a message, the user must determine whether the certificate has been revoked. The user could check the directory each time a certificate is received. To avoid the delays (and possible costs) associated with directory searches, it is likely that the user would maintain a local cache of certificates and lists of revoked certificates.

X.509 Version 3

The X.509 version 2 format does not convey all of the information that recent design and implementation experience has shown to be needed. [FORD95] lists the follow- ing requirements not satisfied by version 2.

1. The subject field is inadequate to convey the identity of a key owner to a public-key user. X.509 names may be relatively short and lacking in obvious identification details that may be needed by the user.

2. The subject field is also inadequate for many applications, which typically recognize entities by an Internet email address, a URL, or some other Internet- related identification.

3. There is a need to indicate security policy information. This enables a security application or function, such as IPSec, to relate an X.509 certificate to a given policy.

4. There is a need to limit the damage that can result from a faulty or malicious CA by setting constraints on the applicability of a particular certificate.

5. It is important to be able to identify different keys used by the same owner at different times. This feature supports key lifecycle management: in particular, the ability to update key pairs for users and CAs on a regular basis or under exceptional circumstances.

Rather than continue to add fields to a fixed format, standards developers felt that a more flexible approach was needed. Thus, version 3 includes a number of optional extensions that may be added to the version 2 format. Each extension consists of an extension identifier, a criticality indicator, and an extension value. The criticality indicator indicates whether an extension can be safely ignored. If the indicator has a value of TRUE and an implementation does not recognize the extension, it must treat the certificate as invalid.

The certificate extensions fall into three main categories: key and policy information, subject and issuer attributes, and certification path constraints.

KEY AND POLICY INFORMATION These extensions convey additional information about the subject and issuer keys, plus indicators of certificate policy. A certif- icate policy is a named set of rules that indicates the applicability of a certifi- cate to a particular community and/or class of application with common security requirements. For example, a policy might be applicable to the authentication of

466 CHAPTER 14 / KEY MANAGEMENT AND DISTRIBUTION

electronic data interchange (EDI) transactions for the trading of goods within a given price range.

This area includes:

■ Authority key identifier: Identifies the public key to be used to verify the signature on this certificate or CRL. Enables distinct keys of the same CA to be differentiated. One use of this field is to handle CA key pair updating.

■ Subject key identifier: Identifies the public key being certified. Useful for sub- ject key pair updating. Also, a subject may have multiple key pairs and, cor- respondingly, different certificates for different purposes (e.g., digital signature and encryption key agreement).

■ Key usage: Indicates a restriction imposed as to the purposes for which, and the policies under which, the certified public key may be used. May indicate one or more of the following: digital signature, nonrepudiation, key encryp- tion, data encryption, key agreement, CA signature verification on certificates, CA signature verification on CRLs.

■ Private-key usage period: Indicates the period of use of the private key cor- responding to the public key. Typically, the private key is used over a different period from the validity of the public key. For example, with digital signature keys, the usage period for the signing private key is typically shorter than that for the verifying public key.

■ Certificate policies: Certificates may be used in environments where multiple policies apply. This extension lists policies that the certificate is recognized as supporting, together with optional qualifier information.

■ Policy mappings: Used only in certificates for CAs issued by other CAs. Policy mappings allow an issuing CA to indicate that one or more of that issuer’s policies can be considered equivalent to another policy used in the subject CA’s domain.

CERTIFICATE SUBJECT AND ISSUER ATTRIBUTES These extensions support alterna- tive names, in alternative formats, for a certificate subject or certificate issuer and can convey additional information about the certificate subject to increase a cer- tificate user’s confidence that the certificate subject is a particular person or entity. For example, information such as postal address, position within a corporation, or picture image may be required.

The extension fields in this area include:

■ Subject alternative name: Contains one or more alternative names, using any of a variety of forms. This field is important for supporting certain applications, such as electronic mail, EDI, and IPSec, which may employ their own name forms.

■ Issuer alternative name: Contains one or more alternative names, using any of a variety of forms.

■ Subject directory attributes: Conveys any desired X.500 directory attribute values for the subject of this certificate.

14.5 / PUBLIC-KEY INFRASTRUCTURE 467

CERTIFICATION PATH CONSTRAINTS These extensions allow constraint specifications to be included in certificates issued for CAs by other CAs. The constraints may restrict the types of certificates that can be issued by the subject CA or that may occur subsequently in a certification chain.

The extension fields in this area include:

■ Basic constraints: Indicates if the subject may act as a CA. If so, a certification path length constraint may be specified.

■ Name constraints: Indicates a name space within which all subject names in subsequent certificates in a certification path must be located.

■ Policy constraints: Specifies constraints that may require explicit certifi- cate policy identification or inhibit policy mapping for the remainder of the certification path.

14.5 PUBLIC-KEY INFRASTRUCTURE

RFC 4949 (Internet Security Glossary) defines public-key infrastructure (PKI) as the set of hardware, software, people, policies, and procedures needed to create, manage, store, distribute, and revoke digital certificates based on asymmetric cryptography. The principal objective for developing a PKI is to enable secure, convenient, and efficient acquisition of public keys. The Internet Engineering Task Force (IETF) Public Key Infrastructure X.509 (PKIX) working group has been the driving force behind setting up a formal (and generic) model based on X.509 that is suitable for deploying a certificate-based architecture on the Internet. This section describes the PKIX model.

Figure 14.17 shows the interrelationship among the key elements of the PKIX model. These elements are

■ End entity: A generic term used to denote end users, devices (e.g., servers, routers), or any other entity that can be identified in the subject field of a public-key certificate. End entities typically consume and/or support PKI- related services.

■ Certification authority (CA): The issuer of certificates and (usually) certifi- cate revocation lists (CRLs). It may also support a variety of administrative functions, although these are often delegated to one or more Registration Authorities.

■ Registration authority (RA): An optional component that can assume a num- ber of administrative functions from the CA. The RA is often associated with the end entity registration process but can assist in a number of other areas as well.

■ CRL issuer: An optional component that a CA can delegate to publish CRLs.

■ Repository: A generic term used to denote any method for storing certificates and CRLs so that they can be retrieved by end entities.

468 CHAPTER 14 / KEY MANAGEMENT AND DISTRIBUTION

PKIX Management Functions

PKIX identifies a number of management functions that potentially need to be supported by management protocols. These are indicated in Figure 14.17 and include the following:

■ Registration: This is the process whereby a user first makes itself known to a CA (directly or through an RA), prior to that CA issuing a certificate or certificates for that user. Registration begins the process of enrolling in a PKI. Registration usually involves some offline or online procedure for mutual authentication. Typically, the end entity is issued one or more shared secret keys used for subsequent authentication.

■ Initialization: Before a client system can operate securely, it is necessary to install key materials that have the appropriate relationship with keys stored elsewhere in the infrastructure. For example, the client needs to be securely initialized with the public key and other assured information of the trusted CA(s), to be used in validating certificate paths.

■ Certification: This is the process in which a CA issues a certificate for a user’s public key, returns that certificate to the user’s client system, and/or posts that certificate in a repository.

■ Key pair recovery: Key pairs can be used to support digital signature creation and verification, encryption and decryption, or both. When a key pair is used for

Figure 14.17 PKIX Architectural Model

End entity Certificate/CRL retrieval

Certificate publication

Certificate/CRL publication

CRL publication

Cross certification

C er

tifi ca

te /C

R L

R ep

os ito

ry

Certificate authority

Registration authority

Certificate authority

Registration, initialization, certification, key pair recovery, key pair update revocation request

PKI users

PKI management

entities

CRL issuer

14.6 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 469

encryption/decryption, it is important to provide a mechanism to recover the necessary decryption keys when normal access to the keying material is no longer possible, otherwise it will not be possible to recover the encrypted data. Loss of access to the decryption key can result from forgotten passwords/PINs, corrupted disk drives, damage to hardware tokens, and so on. Key pair recovery allows end entities to restore their encryption/decryption key pair from an authorized key backup facility (typically, the CA that issued the end entity’s certificate).

■ Key pair update: All key pairs need to be updated regularly (i.e., replaced with a new key pair) and new certificates issued. Update is required when the certificate lifetime expires and as a result of certificate revocation.

■ Revocation request: An authorized person advises a CA of an abnormal situ- ation requiring certificate revocation. Reasons for revocation include private- key compromise, change in affiliation, and name change.

■ Cross certification: Two CAs exchange information used in establishing a cross-certificate. A cross-certificate is a certificate issued by one CA to another CA that contains a CA signature key used for issuing certificates.

PKIX Management Protocols

The PKIX working group has defines two alternative management protocols between PKIX entities that support the management functions listed in the pre- ceding subsection. RFC 2510 defines the certificate management protocols (CMP). Within CMP, each of the management functions is explicitly identified by specific protocol exchanges. CMP is designed to be a flexible protocol able to accommodate a variety of technical, operational, and business models.

RFC 2797 defines certificate management messages over CMS (CMC), where CMS refers to RFC 2630, cryptographic message syntax. CMC is built on earlier work and is intended to leverage existing implementations. Although all of the PKIX func- tions are supported, the functions do not all map into specific protocol exchanges.

14.6 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS

Key Terms

Review Questions 14.1 Explain why man-in-the-middle attacks are ineffective on the secret key distribution

protocol discussed in Figure 14.3. 14.2 What is the major issue in end to end key distribution? How does the key hierarchy

concept address that issue? 14.3 What is a nonce? 14.4 What is a key distribution center? 14.5 What are two different uses of public-key cryptography related to key distribution?

end-to-end encryption key distribution key distribution center (KDC) key management

man-in-the-middle attack master key nonce public-key certificate

public-key directory X.509 certificate

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470 CHAPTER 14 / KEY MANAGEMENT AND DISTRIBUTION

14.6 List four general categories of schemes for the distribution of public keys. 14.7 Discuss the potential security issues that arise due to public key directory based

system. 14.8 What is a public-key certificate? 14.9 What are the requirements for the use of a public-key certificate scheme? 14.10 What is the purpose of the X.509 standard? 14.11 What is a chain of certificates? 14.12 How is an X.509 certificate revoked?

Problems 14.1 One local area network vendor provides a key distribution facility, as illustrated in

Figure 14.18. a. Describe the scheme. b. Compare this scheme to that of Figure 14.3. What are the pros and cons?

14.2 “We are under great pressure, Holmes.” Detective Lestrade looked nervous. “We have learned that copies of sensitive government documents are stored in computers of one foreign embassy here in London. Normally these documents exist in electronic form only on a selected few government computers that satisfy the most stringent security requirements. However, sometimes they must be sent through the network connecting all government computers. But all messages in this network are encrypted using a top-secret encryption algorithm certified by our best crypto experts. Even the NSA and the KGB are unable to break it. And now these documents have appeared in hands of diplomats of a small, otherwise insignificant, country. And we have no idea how it could happen.”

“But you do have some suspicion who did it, do you?” asked Holmes. “Yes, we did some routine investigation. There is a man who has legal access

to one of the government computers and has frequent contacts with diplomats from the embassy. But the computer he has access to is not one of the trusted ones where these documents are normally stored. He is the suspect, but we have no idea how he could obtain copies of the documents. Even if he could obtain a copy of an encrypted document, he couldn’t decrypt it.”

Figure 14.18 Figure for Problem 14.1

Key Distribution

Center (KDC)

B A

(1) IDA, E(Ka, Na)

(2) IDA, E(Ka, Na), IDB, E(Kb, Nb)

(4) E(Ka, [Ks, IDB, Na])

(3) E(Kb, [Ks, IDA, Nb]), E(Ka, [Ks, IDB, Na])

14.6 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 471

“Hmm, please describe the communication protocol used on the network.” Holmes opened his eyes, thus proving that he had followed Lestrade’s talk with an attention that contrasted with his sleepy look.

“Well, the protocol is as follows. Each node N of the network has been assigned a unique secret key Kn. This key is used to secure communication between the node and a trusted server. That is, all the keys are stored also on the server. User A, wishing to send a secret message M to user B, initiates the following protocol:

1. A generates a random number R and sends to the server his name A, destination B, and E(Ka, R).

2. Server responds by sending E(Kb, R) to A. 3. A sends E(R, M) together with E(Kb, R) to B. 4. B knows Kb, thus decrypts E(Kb, R), to get R and will subsequently use R to

decrypt E(R, M) to get M.

You see that a random key is generated every time a message has to be sent. I admit the man could intercept messages sent between the top-secret trusted nodes, but I see no way he could decrypt them.”

“Well, I think you have your man, Lestrade. The protocol isn’t secure because the server doesn’t authenticate users who send him a request. Apparently designers of the protocol have believed that sending E(Kx, R) implicitly authenticates user X as the sender, as only X (and the server) knows Kx. But you know that E(Kx, R) can be intercepted and later replayed. Once you understand where the hole is, you will be able to obtain enough evidence by monitoring the man’s use of the computer he has access to. Most likely he works as follows. After intercepting E(Ka, R) and E(R, M) (see steps 1 and 3 of the protocol), the man, let’s denote him as Z, will continue by pretending to be A and . . .

Finish the sentence for Holmes. 14.3 The 1988 version of X.509 lists properties that RSA keys must satisfy to be secure

given current knowledge about the difficulty of factoring large numbers. The discus- sion concludes with a constraint on the public exponent and the modulus n:

It must be ensured that e 7 log2(n) to prevent attack by taking the eth root mod n to disclose the plaintext.

Although the constraint is correct, the reason given for requiring it is incorrect. What is wrong with the reason given and what is the correct reason?

14.4 Find at least one intermediate certification authority’s certificate and one trusted root certification authority’s certificate on your computer (e.g., in the browser). Print screenshots of both the general and details tab for each certificate.

14.5 NIST defines the term cryptoperiod as the time span during which a specific key is authorized for use or in which the keys for a given system or application may remain in effect. One document on key management uses the following time diagram for a shared secret key.

Originator usage period

Recipient usage period

Cryptoperiod

472 CHAPTER 14 / KEY MANAGEMENT AND DISTRIBUTION

Explain the overlap by giving an example application in which the originator’s usage period for the shared secret key begins before the recipient’s usage period and also ends before the recipients usage period.

14.6 Consider the following protocol, designed to let A and B decide on a fresh, shared session key KAB

= . We assume that they already share a long-term key KAB. 1. A S B: A, NA. 2. B S A: E(KAB, [NA, KAB= ]) 3. A S B: E(KAB= , NA) a. We first try to understand the protocol designer’s reasoning: —Why would A and B believe after the protocol ran that they share KAB

= with the other party?

—Why would they believe that this shared key is fresh? In both cases, you should explain both the reasons of both A and B, so your answer should complete the sentences A believes that she shares KAB

= with B since . . . B believes that he shares KAB

= with A since . . . A believes that KAB

= is fresh since . . . B believes that KAB

= is fresh since . . . b. Assume now that A starts a run of this protocol with B. However, the connection

is intercepted by the adversary C. Show how C can start a new run of the protocol using reflection, causing A to believe that she has agreed on a fresh key with B (in spite of the fact that she has only been communicating with C). Thus, in particular, the belief in (a) is false.

c. Propose a modification of the protocol that prevents this attack. 14.7 What are the management functions of a PKI? What is a cross certificate? 14.8 State the significance of key pair recovery. When is the key pair updated?

Note: The remaining problems deal with the a cryptographic product developed by IBM, which is briefly described in a document at box.com/Crypto7e (IBMCrypto.pdf). Try these problems after reviewing the document.

14.9 What is the effect of adding the instruction EMKi

EMKi: X S E(KMHi, X) i = 0, 1

14.10 Suppose N different systems use the IBM Cryptographic Subsystem with host master keys KMH[i](i = 1, 2, c N). Devise a method for communicating between sys- tems without requiring the system to either share a common host master key or to divulge their individual host master keys. Hint: Each system needs three variants of its host master key.

14.11 The principal objective of the IBM Cryptographic Subsystem is to protect transmis- sions between a terminal and the processing system. Devise a procedure, perhaps adding instructions, which will allow the processor to generate a session key KS and distribute it to Terminal i and Terminal j without having to store a key-equivalent variable in the host.

473

CHAPTER

User Authentication 15.1 Remote User-Authentication Principles

The NIST Model for Electronic User Authentication Means of Authentication Mutual Authentication One-Way Authentication

15.2 Remote User-Authentication Using Symmetric Encryption

Mutual Authentication One-Way Authentication

15.3 Kerberos

Motivation Kerberos Version 4 Kerberos Version 5

15.4 Remote User-Authentication Using Asymmetric Encryption

Mutual Authentication One-Way Authentication

15.5 Federated Identity Management

Identity Management Identity Federation

15.6 Personal Identity Verification

PIV System Model PIV Documentation PIV Credentials and Keys Authentication

15.7 Key Terms, Review Questions, and Problems

474 CHAPTER 15 / USER AUTHENTICATION

This chapter examines some of the authentication functions that have been developed to support network-based user authentication. The chapter begins with an introduc- tion to some of the concepts and key considerations for user authentication over a network or the Internet. The next section examines user-authentication protocols that rely on symmetric encryption. This is followed by a section on one of the earliest and also one of the most widely used authentication services: Kerberos. Next, the chapter looks at user-authentication protocols that rely on asymmetric encryption. This is fol- lowed by a discussion of the X.509 user-authentication protocol. Finally, the concept of federated identity is introduced.

15.1 REMOTE USER-AUTHENTICATION PRINCIPLES

In most computer security contexts, user authentication is the fundamental build- ing block and the primary line of defense. User authentication is the basis for most types of access control and for user accountability. RFC 4949 (Internet Security Glossary) defines user authentication as the process of verifying an identity claimed by or for a system entity. This process consists of two steps:

■ Identification step: Presenting an identifier to the security system. (Identifiers should be assigned carefully, because authenticated identities are the basis for other security services, such as access control service.)

■ Verification step: Presenting or generating authentication information that corroborates the binding between the entity and the identifier.

For example, user Alice Toklas could have the user identifier ABTOKLAS. This information needs to be stored on any server or computer system that Alice wishes to use and could be known to system administrators and other users.

LEARNING OBJECTIVES

After studying this chapter, you should be able to:

◆ Understand the distinction between identification and verification.

◆ Present an overview of techniques for remote user authentication using symmetric encryption.

◆ Give a presentation on Kerberos.

◆ Explain the differences between versions 4 and 5 of Kerberos.

◆ Describe the use of Kerberos in multiple realms.

◆ Present an overview of techniques for remote user authentication using asymmetric encryption.

◆ Understand the need for a federated identity management system.

◆ Explain the use of PIV mechanisms as part of a user authentication system.

15.1 / REMOTE USER-AUTHENTICATION PRINCIPLES 475

A typical item of authentication information associated with this user ID is a pass- word, which is kept secret (known only to Alice and to the system). If no one is able to obtain or guess Alice’s password, then the combination of Alice’s user ID and password enables administrators to set up Alice’s access permissions and audit her activity. Because Alice’s ID is not secret, system users can send her email, but because her password is secret, no one can pretend to be Alice.

In essence, identification is the means by which a user provides a claimed identity to the system; user authentication is the means of establishing the validity of the claim. Note that user authentication is distinct from message authentication. As defined in Chapter 12, message authentication is a procedure that allows com- municating parties to verify that the contents of a received message have not been altered and that the source is authentic. This chapter is concerned solely with user authentication.

The NIST Model for Electronic User Authentication

NIST SP 800-63-2 (Electronic Authentication Guideline, August 2013) defines elec- tronic user authentication as the process of establishing confidence in user identi- ties that are presented electronically to an information system. Systems can use the authenticated identity to determine if the authenticated individual is authorized to perform particular functions, such as database transactions or access to system re- sources. In many cases, the authentication and transaction or other authorized function takes place across an open network such as the Internet. Equally authentication and subsequent authorization can take place locally, such as across a local area network.

SP 800-63-2 defines a general model for user authentication that involves a num- ber of entities and procedures. We discuss this model with reference to Figure 15.1.

The initial requirement for performing user authentication is that the user must be registered with the system. The following is a typical sequence for registra- tion. An applicant applies to a registration authority (RA) to become a subscriber

Figure 15.1 The NIST SP 800-63-2 E-Authentication Architectural Model

Registration authority (RA)

Registration, credential issuance, and maintenance

E-Authentication using token and credential

Identity proofing User registration

To ken

, cr ede

nti al

Re gis

tra tio

n/i ssu

anc e

Authenticated session

Authenticated protocol

Exchange

Authenticated assertion

Registration Confirmation

Token/credential Validation

Relying party (RP)

Verifier

Subscriber/ claimant

Credential service

provider (RA)

476 CHAPTER 15 / USER AUTHENTICATION

of a credential service provider (CSP). In this model, the RA is a trusted entity that establishes and vouches for the identity of an applicant to a CSP. The CSP then engages in an exchange with the subscriber. Depending on the details of the over- all authentication system, the CSP issues some sort of electronic credential to the subscriber. The credential is a data structure that authoritatively binds an identity and additional attributes to a token possessed by a subscriber, and can be verified when presented to the verifier in an authentication transaction. The token could be an encryption key or an encrypted password that identifies the subscriber. The token may be issued by the CSP, generated directly by the subscriber, or provided by a third party. The token and credential may be used in subsequent authentica- tion events.

Once a user is registered as a subscriber, the actual authentication process can take place between the subscriber and one or more systems that perform authen- tication and, subsequently, authorization. The party to be authenticated is called a claimant and the party verifying that identity is called a verifier. When a claimant successfully demonstrates possession and control of a token to a verifier through an authentication protocol, the verifier can verify that the claimant is the subscriber named in the corresponding credential. The verifier passes on an assertion about the identity of the subscriber to the relying party (RP). That assertion includes identity information about a subscriber, such as the subscriber name, an identifier assigned at registration, or other subscriber attributes that were verified in the registration process. The RP can use the authenticated information provided by the verifier to make access control or authorization decisions.

An implemented system for authentication will differ from or be more com- plex than this simplified model, but the model illustrates the key roles and functions needed for a secure authentication system.

Means of Authentication

There are four general means of authenticating a user’s identity, which can be used alone or in combination:

■ Something the individual knows: Examples include a password, a personal identification number (PIN), or answers to a prearranged set of questions.

■ Something the individual possesses: Examples include cryptographic keys, electronic keycards, smart cards, and physical keys. This type of authenticator is referred to as a token.

■ Something the individual is (static biometrics): Examples include recognition by fingerprint, retina, and face.

■ Something the individual does (dynamic biometrics): Examples include recog- nition by voice pattern, handwriting characteristics, and typing rhythm.

All of these methods, properly implemented and used, can provide secure user authentication. However, each method has problems. An adversary may be able to guess or steal a password. Similarly, an adversary may be able to forge or steal a token. A user may forget a password or lose a token. Furthermore, there is a significant administrative overhead for managing password and token information on systems and securing such information on systems. With respect to biometric

15.1 / REMOTE USER-AUTHENTICATION PRINCIPLES 477

authenticators, there are a variety of problems, including dealing with false positives and false negatives, user acceptance, cost, and convenience. For network-based user authentication, the most important methods involve cryptographic keys and some- thing the individual knows, such as a password.

Mutual Authentication

An important application area is that of mutual authentication protocols. Such pro- tocols enable communicating parties to satisfy themselves mutually about each oth- er’s identity and to exchange session keys. This topic was examined in Chapter 14. There, the focus was key distribution. We return to this topic here to consider the wider implications of authentication.

Central to the problem of authenticated key exchange are two issues: confi- dentiality and timeliness. To prevent masquerade and to prevent compromise of session keys, essential identification and session-key information must be commu- nicated in encrypted form. This requires the prior existence of secret or public keys that can be used for this purpose. The second issue, timeliness, is important because of the threat of message replays. Such replays, at worst, could allow an opponent to compromise a session key or successfully impersonate another party. At minimum, a successful replay can disrupt operations by presenting parties with messages that appear genuine but are not.

[GONG93] lists the following examples of replay attacks:

1. The simplest replay attack is one in which the opponent simply copies a mes- sage and replays it later.

2. An opponent can replay a timestamped message within the valid time window. If both the original and the replay arrive within then time window, this inci- dent can be logged.

3. As with example (2), an opponent can replay a timestamped message within the valid time window, but in addition, the opponent suppresses the original message. Thus, the repetition cannot be detected.

4. Another attack involves a backward replay without modification. This is a re- play back to the message sender. This attack is possible if symmetric encryp- tion is used and the sender cannot easily recognize the difference between messages sent and messages received on the basis of content.

One approach to coping with replay attacks is to attach a sequence number to each message used in an authentication exchange. A new message is accepted only if its sequence number is in the proper order. The difficulty with this approach is that it requires each party to keep track of the last sequence number for each claim- ant it has dealt with. Because of this overhead, sequence numbers are generally not used for authentication and key exchange. Instead, one of the following two general approaches is used:

■ Timestamps: Party A accepts a message as fresh only if the message contains a timestamp that, in A’s judgment, is close enough to A’s knowledge of cur- rent time. This approach requires that clocks among the various participants be synchronized.

478 CHAPTER 15 / USER AUTHENTICATION

■ Challenge/response: Party A, expecting a fresh message from B, first sends B a nonce (challenge) and requires that the subsequent message (response) re- ceived from B contain the correct nonce value.

It can be argued (e.g., [LAM92a]) that the timestamp approach should not be used for connection-oriented applications because of the inherent difficulties with this technique. First, some sort of protocol is needed to maintain synchronization among the various processor clocks. This protocol must be both fault tolerant, to cope with network errors, and secure, to cope with hostile attacks. Second, the oppor- tunity for a successful attack will arise if there is a temporary loss of synchronization resulting from a fault in the clock mechanism of one of the parties. Finally, because of the variable and unpredictable nature of network delays, distributed clocks cannot be expected to maintain precise synchronization. Therefore, any timestamp-based procedure must allow for a window of time sufficiently large to accommodate net- work delays yet sufficiently small to minimize the opportunity for attack.

On the other hand, the challenge-response approach is unsuitable for a con- nectionless type of application, because it requires the overhead of a handshake be- fore any connectionless transmission, effectively negating the chief characteristic of a connectionless transaction. For such applications, reliance on some sort of secure time server and a consistent attempt by each party to keep its clocks in synchroniza- tion may be the best approach (e.g., [LAM92b]).

One-Way Authentication

One application for which encryption is growing in popularity is electronic mail (email). The very nature of electronic mail, and its chief benefit, is that it is not nec- essary for the sender and receiver to be online at the same time. Instead, the email message is forwarded to the receiver’s electronic mailbox, where it is buffered until the receiver is available to read it.

The “envelope” or header of the email message must be in the clear, so that the message can be handled by the store-and-forward email protocol, such as the Simple Mail Transfer Protocol (SMTP) or X.400. However, it is often desirable that the mail-handling protocol not require access to the plaintext form of the message, because that would require trusting the mail-handling mechanism. Accordingly, the email message should be encrypted such that the mail-handling system is not in possession of the decryption key.

A second requirement is that of authentication. Typically, the recipient wants some assurance that the message is from the alleged sender.

15.2 REMOTE USER-AUTHENTICATION USING SYMMETRIC ENCRYPTION

Mutual Authentication

As was discussed in Chapter 14, a two-level hierarchy of symmetric encryption keys can be used to provide confidentiality for communication in a distributed environ- ment. In general, this strategy involves the use of a trusted key distribution center

Hiva-Network.Com

http://www.hiva-network.com/

15.2 / REMOTE USER-AUTHENTICATION USING SYMMETRIC ENCRYPTION 479

(KDC). Each party in the network shares a secret key, known as a master key, with the KDC. The KDC is responsible for generating keys to be used for a short time over a connection between two parties, known as session keys, and for distribut- ing those keys using the master keys to protect the distribution. This approach is quite common. As an example, we look at the Kerberos system in Section 15.3. The discussion in this subsection is relevant to an understanding of the Kerberos mechanisms.

Figure 14.3 illustrates a proposal initially put forth by Needham and Schroeder [NEED78] for secret key distribution using a KDC that, as was mentioned in Chapter 14, includes authentication features. The protocol can be summarized as follows.1

1. A S KDC: IDA } IDB }N1 2. KDC S A: E(Ka, [Ks } IDB }N1 }E(Kb, [Ks } IDA])]) 3. A S B: E(Kb, [Ks } IDA]) 4. B S A: E(Ks, N2) 5. A S B: E(Ks, f(N2)) where f() is a generic function that modifies the

value of the nonce.

Secret keys Ka and Kb are shared between A and the KDC and B and the KDC, respectively. The purpose of the protocol is to distribute securely a session key Ks to A and B. Entity A securely acquires a new session key in step 2. The mes- sage in step 3 can be decrypted, and hence understood, only by B. Step 4 reflects B’s knowledge of Ks, and step 5 assures B of A’s knowledge of Ks and assures B that this is a fresh message because of the use of the nonce N2. Recall from our discussion in Chapter 14 that the purpose of steps 4 and 5 is to prevent a certain type of replay at- tack. In particular, if an opponent is able to capture the message in step 3 and replay it, this might in some fashion disrupt operations at B.

Despite the handshake of steps 4 and 5, the protocol is still vulnerable to a form of replay attack. Suppose that an opponent, X, has been able to compromise an old session key. Admittedly, this is a much more unlikely occurrence than that an opponent has simply observed and recorded step 3. Nevertheless, it is a potential security risk. X can impersonate A and trick B into using the old key by simply re- playing step 3. Unless B remembers indefinitely all previous session keys used with A, B will be unable to determine that this is a replay. If X can intercept the hand- shake message in step 4, then it can impersonate A’s response in step 5. From this point on, X can send bogus messages to B that appear to B to come from A using an authenticated session key.

Denning [DENN81, DENN82] proposes to overcome this weakness by a modification to the Needham/Schroeder protocol that includes the addition of a timestamp to steps 2 and 3. Her proposal assumes that the master keys, Ka and Kb, are secure, and it consists of the following steps.

1The portion to the left of the colon indicates the sender and the receiver; the portion to the right indi- cates the contents of the message; the symbol } indicates concatenation.

480 CHAPTER 15 / USER AUTHENTICATION

1. A S KDC: IDA } IDB 2. KDC S A: E(Ka, [Ks } IDB }T }E(Kb, [Ks } IDA }T])]) 3. A S B: E(Kb, [Ks } IDA }T]) 4. B S A: E(Ks, N1) 5. A S B: E(Ks, f(N1))

T is a timestamp that assures A and B that the session key has only just been generated. Thus, both A and B know that the key distribution is a fresh exchange. A and B can verify timeliness by checking that

�Clock – T � 6 ∆t1 + ∆t2 where ∆t1 is the estimated normal discrepancy between the KDC’s clock and the local clock (at A or B) and ∆t2 is the expected network delay time. Each node can set its clock against some standard reference source. Because the timestamp T is encrypted using the secure master keys, an opponent, even with knowledge of an old session key, cannot succeed because a replay of step 3 will be detected by B as untimely.

A final point: Steps 4 and 5 were not included in the original presentation [DENN81] but were added later [DENN82]. These steps confirm the receipt of the session key at B.

The Denning protocol seems to provide an increased degree of security com- pared to the Needham/Schroeder protocol. However, a new concern is raised: namely, that this new scheme requires reliance on clocks that are synchronized throughout the network. [GONG92] points out a risk involved. The risk is based on the fact that the distributed clocks can become unsynchronized as a result of sabotage on or faults in the clocks or the synchronization mechanism.2 The problem occurs when a sender’s clock is ahead of the intended recipient’s clock. In this case, an opponent can intercept a message from the sender and replay it later when the timestamp in the message becomes current at the recipient’s site. This replay could cause unexpected results. Gong refers to such attacks as suppress-replay attacks.

One way to counter suppress-replay attacks is to enforce the requirement that parties regularly check their clocks against the KDC’s clock. The other alternative, which avoids the need for clock synchronization, is to rely on handshaking protocols using nonces. This latter alternative is not vulnerable to a suppress-replay attack, because the nonces the recipient will choose in the future are unpredictable to the sender. The Needham/Schroeder protocol relies on nonces only but, as we have seen, has other vulnerabilities.

In [KEHN92], an attempt is made to respond to the concerns about suppress- replay attacks and at the same time fix the problems in the Needham/Schroeder protocol. Subsequently, an inconsistency in this latter protocol was noted and an improved strategy was presented in [NEUM93a].3 The protocol is

2Such things can and do happen. In recent years, flawed chips were used in a number of computers and other electronic systems to track the time and date. The chips had a tendency to skip forward one day. [NEUM90] 3It really is hard to get these things right.

15.2 / REMOTE USER-AUTHENTICATION USING SYMMETRIC ENCRYPTION 481

1. A S B: IDA }Na 2. B S KDC: IDB }Nb }E(Kb, [IDA }Na }Tb]) 3. KDC S A: E(Ka, [IDB }Na }Ks }Tb]) }E(Kb, [IDA }Ks }Tb]) }Nb 4. A S B: E(Kb, [IDA }Ks }Tb]) }E(Ks, Nb)

Let us follow this exchange step by step.

1. A initiates the authentication exchange by generating a nonce, Na, and sending that plus its identifier to B in plaintext. This nonce will be returned to A in an encrypted message that includes the session key, assuring A of its timeliness.

2. B alerts the KDC that a session key is needed. Its message to the KDC in- cludes its identifier and a nonce, Nb. This nonce will be returned to B in an encrypted message that includes the session key, assuring B of its timeliness. B’s message to the KDC also includes a block encrypted with the secret key shared by B and the KDC. This block is used to instruct the KDC to issue credentials to A; the block specifies the intended recipient of the credentials, a suggested expiration time for the credentials, and the nonce received from A.

3. The KDC passes on to A B’s nonce and a block encrypted with the secret key that B shares with the KDC. The block serves as a “ticket” that can be used by A for subsequent authentications, as will be seen. The KDC also sends to A a block encrypted with the secret key shared by A and the KDC. This block verifies that B has received A’s initial message (IDB) and that this is a timely message and not a replay (Na), and it provides A with a session key (Ks) and the time limit on its use (Tb).

4. A transmits the ticket to B, together with the B’s nonce, the latter encrypted with the session key. The ticket provides B with the secret key that is used to de- crypt E(Ks, Nb) to recover the nonce. The fact that B’s nonce is encrypted with the session key authenticates that the message came from A and is not a replay.

This protocol provides an effective, secure means for A and B to establish a session with a secure session key. Furthermore, the protocol leaves A in posses- sion of a key that can be used for subsequent authentication to B, avoiding the need to contact the authentication server repeatedly. Suppose that A and B estab- lish a session using the aforementioned protocol and then conclude that session. Subsequently, but within the time limit established by the protocol, A desires a new session with B. The following protocol ensues:

1. A S B: E(Kb, [IDA }Ks }Tb]) }Na=

2. B S A: Nb= }E(Ks, Na= )

3. A S B: E(Ks, Nb= )

When B receives the message in step 1, it verifies that the ticket has not expired. The newly generated nonces Na

= and Nb = assure each party that there is no replay

attack. In all the foregoing, the time specified in Tb is a time relative to B’s clock.

Thus, this timestamp does not require synchronized clocks, because B checks only self-generated timestamps.

482 CHAPTER 15 / USER AUTHENTICATION

One-Way Authentication

Using symmetric encryption, the decentralized key distribution scenario illustrated in Figure 14.5 is impractical. This scheme requires the sender to issue a request to the intended recipient, await a response that includes a session key, and only then send the message.

With some refinement, the KDC strategy illustrated in Figure 14.3 is a can- didate for encrypted electronic mail. Because we wish to avoid requiring that the recipient (B) be on line at the same time as the sender (A), steps 4 and 5 must be eliminated. For a message with content M, the sequence is as follows:

1. A S KDC: IDA } IDB }N1 2. KDC S A: E(Ka, [Ks } IDB }N1 }E(Kb, [Ks } IDA])]) 3. A S B: E(Kb, [Ks } IDA]) }E(Ks, M)

This approach guarantees that only the intended recipient of a message will be able to read it. It also provides a level of authentication that the sender is A. As specified, the protocol does not protect against replays. Some measure of defense could be provided by including a timestamp with the message. However, because of the potential delays in the email process, such timestamps may have limited usefulness.

15.3 KERBEROS

Kerberos4 is an authentication service developed as part of Project Athena at MIT. The problem that Kerberos addresses is this: Assume an open distributed environ- ment in which users at workstations wish to access services on servers distributed throughout the network. We would like for servers to be able to restrict access to authorized users and to be able to authenticate requests for service. In this envi- ronment, a workstation cannot be trusted to identify its users correctly to network services. In particular, the following three threats exist:

1. A user may gain access to a particular workstation and pretend to be another user operating from that workstation.

2. A user may alter the network address of a workstation so that the requests sent from the altered workstation appear to come from the impersonated workstation.

3. A user may eavesdrop on exchanges and use a replay attack to gain entrance to a server or to disrupt operations.

In any of these cases, an unauthorized user may be able to gain access to services and data that he or she is not authorized to access. Rather than building in elaborate

4“In Greek mythology, a many headed dog, commonly three, perhaps with a serpent’s tail, the guardian of the entrance of Hades.” From Dictionary of Subjects and Symbols in Art, by James Hall, Harper & Row, 1979. Just as the Greek Kerberos has three heads, the modern Kerberos was intended to have three components to guard a network’s gate: authentication, accounting, and audit. The last two heads were never implemented.

15.3 / KERBEROS 483

authentication protocols at each server, Kerberos provides a centralized authenti- cation server whose function is to authenticate users to servers and servers to users. Unlike most other authentication schemes described in this book, Kerberos relies exclusively on symmetric encryption, making no use of public-key encryption.

Two versions of Kerberos are in common use. Version 4 [MILL88, STEI88] implementations still exist. Version 5 [KOHL94] corrects some of the security defi- ciencies of version 4 and has been issued as a proposed Internet Standard (RFC 4120 and RFC 4121).5

We begin this section with a brief discussion of the motivation for the Kerberos approach. Then, because of the complexity of Kerberos, it is best to start with a de- scription of the authentication protocol used in version 4. This enables us to see the essence of the Kerberos strategy without considering some of the details required to handle subtle security threats. Finally, we examine version 5.

Motivation

If a set of users is provided with dedicated personal computers that have no network connections, then a user’s resources and files can be protected by physically secur- ing each personal computer. When these users instead are served by a centralized time-sharing system, the time-sharing operating system must provide the security. The operating system can enforce access-control policies based on user identity and use the logon procedure to identify users.

Today, neither of these scenarios is typical. More common is a distributed architecture consisting of dedicated user workstations (clients) and distributed or centralized servers. In this environment, three approaches to security can be envisioned.

1. Rely on each individual client workstation to assure the identity of its user or users and rely on each server to enforce a security policy based on user iden- tification (ID).

2. Require that client systems authenticate themselves to servers, but trust the client system concerning the identity of its user.

3. Require the user to prove his or her identity for each service invoked. Also require that servers prove their identity to clients.

In a small, closed environment in which all systems are owned and operated by a single organization, the first or perhaps the second strategy may suffice.6 But in a more open environment in which network connections to other machines are supported, the third approach is needed to protect user information and resources housed at the server. Kerberos supports this third approach. Kerberos assumes a distributed client/server architecture and employs one or more Kerberos servers to provide an authentication service.

5Versions 1 through 3 were internal development versions. Version 4 is the “original” Kerberos. 6However, even a closed environment faces the threat of attack by a disgruntled employee.

484 CHAPTER 15 / USER AUTHENTICATION

The first published report on Kerberos [STEI88] listed the following requirements.

■ Secure: A network eavesdropper should not be able to obtain the necessary information to impersonate a user. More generally, Kerberos should be strong enough that a potential opponent does not find it to be the weak link.

■ Reliable: For all services that rely on Kerberos for access control, lack of availability of the Kerberos service means lack of availability of the supported services. Hence, Kerberos should be highly reliable and should employ a distributed server architecture with one system able to back up another.

■ Transparent: Ideally, the user should not be aware that authentication is taking place beyond the requirement to enter a password.

■ Scalable: The system should be capable of supporting large numbers of clients and servers. This suggests a modular, distributed architecture.

To support these requirements, the overall scheme of Kerberos is that of a trusted third-party authentication service that uses a protocol based on that pro- posed by Needham and Schroeder [NEED78], which was discussed in Section 15.2. It is trusted in the sense that clients and servers trust Kerberos to mediate their mutual authentication. Assuming the Kerberos protocol is well designed, then the authentication service is secure if the Kerberos server itself is secure.7

Kerberos Version 4

Version 4 of Kerberos makes use of DES, in a rather elaborate protocol, to pro- vide the authentication service. Viewing the protocol as a whole, it is difficult to see the need for the many elements contained therein. Therefore, we adopt a strategy used by Bill Bryant of Project Athena [BRYA88] and build up to the full protocol by looking first at several hypothetical dialogues. Each successive dialogue adds additional complexity to counter security vulnerabilities revealed in the preceding dialogue.

After examining the protocol, we look at some other aspects of version 4.

A SIMPLE AUTHENTICATION DIALOGUE In an unprotected network environment, any client can apply to any server for service. The obvious security risk is that of im- personation. An opponent can pretend to be another client and obtain unauthor- ized privileges on server machines. To counter this threat, servers must be able to confirm the identities of clients who request service. Each server can be required to undertake this task for each client/server interaction, but in an open environment, this places a substantial burden on each server.

7Remember that the security of the Kerberos server should not automatically be assumed but must be guarded carefully (e.g., in a locked room). It is well to remember the fate of the Greek Kerberos, whom Hercules was ordered by Eurystheus to capture as his Twelfth Labor: “Hercules found the great dog on its chain and seized it by the throat. At once the three heads tried to attack, and Kerberos lashed about with his powerful tail. Hercules hung on grimly, and Kerberos relaxed into unconsciousness. Eurystheus may have been surprised to see Hercules alive—when he saw the three slavering heads and the huge dog they belonged to he was frightened out of his wits, and leapt back into the safety of his great bronze jar.” From The Hamlyn Concise Dictionary of Greek and Roman Mythology, by Michael Stapleton, Hamlyn, 1982.

15.3 / KERBEROS 485

An alternative is to use an authentication server (AS) that knows the passwords of all users and stores these in a centralized database. In addition, the AS shares a unique secret key with each server. These keys have been distributed physi- cally or in some other secure manner. Consider the following hypothetical dialogue:

(1) C S AS: IDC }PC } IDV (2) AS S C: Ticket (3) C S V: IDC }Ticket

Ticket = E(Kv, [IDC }ADC } IDV])

where

C = client AS = authentication server

V = server IDC = identifier of user on C IDV = identifier of V

PC = password of user on C ADC = network address of C

Kv = secret encryption key shared by AS and V

In this scenario, the user logs on to a workstation and requests access to server V. The client module C in the user’s workstation requests the user’s password and then sends a message to the AS that includes the user’s ID, the server’s ID, and the user’s password. The AS checks its database to see if the user has supplied the proper password for this user ID and whether this user is permitted access to server V. If both tests are passed, the AS accepts the user as authentic and must now convince the server that this user is authentic. To do so, the AS creates a ticket that con- tains the user’s ID and network address and the server’s ID. This ticket is encrypted using the secret key shared by the AS and this server. This ticket is then sent back to C. Because the ticket is encrypted, it cannot be altered by C or by an opponent.

With this ticket, C can now apply to V for service. C sends a message to V con- taining C’s ID and the ticket. V decrypts the ticket and verifies that the user ID in the ticket is the same as the unencrypted user ID in the message. If these two match, the server considers the user authenticated and grants the requested service.

Each of the ingredients of message (3) is significant. The ticket is encrypted to prevent alteration or forgery. The server’s ID (IDV) is included in the ticket so that the server can verify that it has decrypted the ticket properly. IDC is included in the ticket to indicate that this ticket has been issued on behalf of C. Finally, ADC serves to counter the following threat. An opponent could capture the ticket transmitted in message (2), then use the name IDC and transmit a message of form (3) from another workstation. The server would receive a valid ticket that matches the user ID and grant access to the user on that other workstation. To prevent this attack, the AS includes in the ticket the network address from which the original request came. Now the ticket is valid only if it is transmitted from the same workstation that initially requested the ticket.

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A MORE SECURE AUTHENTICATION DIALOGUE Although the foregoing scenario solves some of the problems of authentication in an open network environment, problems remain. Two in particular stand out. First, we would like to minimize the number of times that a user has to enter a password. Suppose each ticket can be used only once. If user C logs on to a workstation in the morning and wishes to check his or her mail at a mail server, C must supply a password to get a ticket for the mail server. If C wishes to check the mail several times during the day, each attempt requires re- entering the password. We can improve matters by saying that tickets are reusable. For a single logon session, the workstation can store the mail server ticket after it is received and use it on behalf of the user for multiple accesses to the mail server.

However, under this scheme, it remains the case that a user would need a new ticket for every different service. If a user wished to access a print server, a mail server, a file server, and so on, the first instance of each access would require a new ticket and hence require the user to enter the password.

The second problem is that the earlier scenario involved a plaintext transmis- sion of the password [message (1)]. An eavesdropper could capture the password and use any service accessible to the victim.

To solve these additional problems, we introduce a scheme for avoiding plain- text passwords and a new server, known as the ticket-granting server (TGS). The new (but still hypothetical) scenario is as follows.

Once per user logon session:

(1) C S AS: IDC } IDtgs (2) AS S C: E(Kc, Tickettgs)

Once per type of service:

(3) C S TGS: IDC } IDV }Tickettgs (4) TGS S C: Ticketv

Once per service session:

(5) C S V: IDC }Ticketv Tickettgs = E(Ktgs, [IDC }ADC } IDtgs }TS1 }Lifetime1])

Ticketv = E(Kv, [IDC }ADC } IDv }TS2 }Lifetime2])

The new service, TGS, issues tickets to users who have been authenticated to AS. Thus, the user first requests a ticket-granting ticket (Tickettgs) from the AS. The client module in the user workstation saves this ticket. Each time the user requires access to a new service, the client applies to the TGS, using the ticket to authenti- cate itself. The TGS then grants a ticket for the particular service. The client saves each service-granting ticket and uses it to authenticate its user to a server each time a particular service is requested. Let us look at the details of this scheme:

1. The client requests a ticket-granting ticket on behalf of the user by sending its user’s ID to the AS, together with the TGS ID, indicating a request to use the TGS service.

15.3 / KERBEROS 487

2. The AS responds with a ticket that is encrypted with a key that is derived from the user’s password (Kc), which is already stored at the AS. When this response arrives at the client, the client prompts the user for his or her password, gen- erates the key, and attempts to decrypt the incoming message. If the correct password is supplied, the ticket is successfully recovered.

Because only the correct user should know the password, only the correct user can recover the ticket. Thus, we have used the password to obtain credentials from Kerberos without having to transmit the password in plaintext. The ticket itself consists of the ID and network address of the user, and the ID of the TGS. This corresponds to the first scenario. The idea is that the client can use this ticket to request multiple service-granting tickets. So the ticket-granting ticket is to be reus- able. However, we do not wish an opponent to be able to capture the ticket and use it. Consider the following scenario: An opponent captures the login ticket and waits until the user has logged off his or her workstation. Then the opponent either gains access to that workstation or configures his workstation with the same network ad- dress as that of the victim. The opponent would be able to reuse the ticket to spoof the TGS. To counter this, the ticket includes a timestamp, indicating the date and time at which the ticket was issued, and a lifetime, indicating the length of time for which the ticket is valid (e.g., eight hours). Thus, the client now has a reusable ticket and need not bother the user for a password for each new service request. Finally, note that the ticket-granting ticket is encrypted with a secret key known only to the AS and the TGS. This prevents alteration of the ticket. The ticket is reencrypted with a key based on the user’s password. This assures that the ticket can be recov- ered only by the correct user, providing the authentication.

Now that the client has a ticket-granting ticket, access to any server can be obtained with steps 3 and 4.

3. The client requests a service-granting ticket on behalf of the user. For this pur- pose, the client transmits a message to the TGS containing the user’s ID, the ID of the desired service, and the ticket-granting ticket.

4. The TGS decrypts the incoming ticket using a key shared only by the AS and the TGS (Ktgs) and verifies the success of the decryption by the presence of its ID. It checks to make sure that the lifetime has not expired. Then it compares the user ID and network address with the incoming information to authenti- cate the user. If the user is permitted access to the server V, the TGS issues a ticket to grant access to the requested service.

The service-granting ticket has the same structure as the ticket-granting ticket. Indeed, because the TGS is a server, we would expect that the same elements are needed to authenticate a client to the TGS and to authenticate a client to an appli- cation server. Again, the ticket contains a timestamp and lifetime. If the user wants access to the same service at a later time, the client can simply use the previously acquired service-granting ticket and need not bother the user for a password. Note that the ticket is encrypted with a secret key (Kv) known only to the TGS and the server, preventing alteration.

Finally, with a particular service-granting ticket, the client can gain access to the corresponding service with step 5.

Hiva-Network.Com

http://www.hiva-network.com/

488 CHAPTER 15 / USER AUTHENTICATION

5. The client requests access to a service on behalf of the user. For this purpose, the client transmits a message to the server containing the user’s ID and the service- granting ticket. The server authenticates by using the contents of the ticket.

This new scenario satisfies the two requirements of only one password query per user session and protection of the user password.

THE VERSION 4 AUTHENTICATION DIALOGUE Although the foregoing scenario en- hances security compared to the first attempt, two additional problems remain. The heart of the first problem is the lifetime associated with the ticket-granting ticket. If this lifetime is very short (e.g., minutes), then the user will be repeatedly asked for a password. If the lifetime is long (e.g., hours), then an opponent has a greater opportunity for replay. An opponent could eavesdrop on the network and capture a copy of the ticket-granting ticket and then wait for the legitimate user to log out. Then the opponent could forge the legitimate user’s network address and send the message of step (3) to the TGS. This would give the opponent unlimited access to the resources and files available to the legitimate user.

Similarly, if an opponent captures a service-granting ticket and uses it before it expires, the opponent has access to the corresponding service.

Thus, we arrive at an additional requirement. A network service (the TGS or an application service) must be able to prove that the person using a ticket is the same person to whom that ticket was issued.

The second problem is that there may be a requirement for servers to authen- ticate themselves to users. Without such authentication, an opponent could sabo- tage the configuration so that messages to a server were directed to another loca- tion. The false server would then be in a position to act as a real server and capture any information from the user and deny the true service to the user.

We examine these problems in turn and refer to Table 15.1, which shows the actual Kerberos protocol. Figure 15.2 provides a simplified overview.

(1) C S AS IDc } IDtgs }TS1 (2) AS S C E(Kc, [Kc, tgs } IDtgs }TS2 }Lifetime2 }Tickettgs])

Tickettgs = E(Ktgs, [Kc, tgs } IDC }ADC } IDtgs }TS2 }Lifetime2])

(a) Authentication Service Exchange to obtain ticket-granting ticket

(3) C S TGS IDv }Tickettgs }Authenticatorc (4) TGS S C E(Kc, tgs, [Kc, v } IDv }TS4 }Ticketv])

Tickettgs = E(Ktgs, [Kc, tgs } IDC }ADC } IDtgs }TS2 }Lifetime2]) Ticketv = E(Kv, [Kc, v } IDC }ADC } IDv }TS4 }Lifetime4])

Authenticatorc = E(Kc, tgs, [IDC }ADC }TS3])

(b) Ticket-Granting Service Exchange to obtain service-granting ticket

(5) C S V Ticketv }Authenticatorc (6) V S C E(Kc,v, [TS5 + 1]) (for mutual authentication)

Ticketv = E(Kv, [Kc, v } IDC }ADC } IDv }TS4 }Lifetime4]) Authenticatorc = E(Kc, v, [IDC }ADC }TS5])

(c) Client/Server Authentication Exchange to obtain service

Table 15.1 Summary of Kerberos Version 4 Message Exchanges

15.3 / KERBEROS 489

First, consider the problem of captured ticket-granting tickets and the need to determine that the ticket presenter is the same as the client for whom the ticket was issued. The threat is that an opponent will steal the ticket and use it before it expires. To get around this problem, let us have the AS provide both the client and the TGS with a secret piece of information in a secure manner. Then the client can prove its identity to the TGS by revealing the secret information—again in a secure manner. An efficient way of accomplishing this is to use an encryption key as the secure information; this is referred to as a session key in Kerberos.

Table 15.1a shows the technique for distributing the session key. As before, the client sends a message to the AS requesting access to the TGS. The AS re- sponds with a message, encrypted with a key derived from the user’s password (Kc), that contains the ticket. The encrypted message also contains a copy of the session key, Kc,tgs, where the subscripts indicate that this is a session key for C and TGS. Because this session key is inside the message encrypted with Kc, only the user’s client can read it. The same session key is included in the ticket, which can be read only by the TGS. Thus, the session key has been securely delivered to both C and the TGS.

Figure 15.2 Overview of Kerberos

Authentication server

Ticket- granting

server (TGS)

Host/ application

server

requ est t

icke t-

gran ting

ticke t

once per user logon session

1. User logs on to workstation and requests service on host

3. Workstation prompts user for password to decrypt incoming message, and then send ticket and authenticator that contains user’s name, network address, and time to TGS.

ticket + ses

sion k ey

request se rvice-

granting ticket

ticket + sess ion key

once per type of service

4. TGS decrypts ticket and authenticator, verifies request, and then creates ticket for requested application server.

Kerberos

5. Workstation sends ticket and authenticator to host.

6. Host verifies that ticket and authenticator match, and then grants access to service. If mutual authentication is required, server returns an authenticator.

request service provide server

authenticator once per service session

2. AS verifies user’s access right in database, and creates ticket-granting ticket and session key. Results are encrypted using key derived from user’s password.

490 CHAPTER 15 / USER AUTHENTICATION

Note that several additional pieces of information have been added to this first phase of the dialogue. Message (1) includes a timestamp, so that the AS knows that the message is timely. Message (2) includes several elements of the ticket in a form accessible to C. This enables C to confirm that this ticket is for the TGS and to learn its expiration time.

Armed with the ticket and the session key, C is ready to approach the TGS. As before, C sends the TGS a message that includes the ticket plus the ID of the requested service [message (3) in Table 15.1b]. In addition, C transmits an authentica- tor, which includes the ID and address of C’s user and a timestamp. Unlike the ticket, which is reusable, the authenticator is intended for use only once and has a very short lifetime. The TGS can decrypt the ticket with the key that it shares with the AS. This ticket indicates that user C has been provided with the session key Kc,tgs. In effect, the ticket says, “Anyone who uses Kc,tgs must be C.” The TGS uses the session key to decrypt the authenticator. The TGS can then check the name and address from the authenticator with that of the ticket and with the network address of the incoming message. If all match, then the TGS is assured that the sender of the ticket is indeed the ticket’s real owner. In effect, the authenticator says, “At time TS3, I hereby use Kc,tgs.” Note that the ticket does not prove anyone’s identity but is a way to distribute keys securely. It is the authenticator that proves the client’s identity. Because the au- thenticator can be used only once and has a short lifetime, the threat of an opponent stealing both the ticket and the authenticator for presentation later is countered.

The reply from the TGS in message (4) follows the form of message (2). The message is encrypted with the session key shared by the TGS and C and includes a session key to be shared between C and the server V, the ID of V, and the time- stamp of the ticket. The ticket itself includes the same session key.

C now has a reusable service-granting ticket for V. When C presents this ticket, as shown in message (5), it also sends an authenticator. The server can decrypt the ticket, recover the session key, and decrypt the authenticator.

If mutual authentication is required, the server can reply as shown in message (6) of Table 15.1. The server returns the value of the timestamp from the authenti- cator, incremented by 1, and encrypted in the session key. C can decrypt this mes- sage to recover the incremented timestamp. Because the message was encrypted by the session key, C is assured that it could have been created only by V. The contents of the message assure C that this is not a replay of an old reply.

Finally, at the conclusion of this process, the client and server share a secret key. This key can be used to encrypt future messages between the two or to ex- change a new random session key for that purpose.

Figure 15.3 illustrates the Kerberos exchanges among the parties. Table 15.2 summarizes the justification for each of the elements in the Kerberos protocol.

KERBEROS REALMS AND MULTIPLE KERBERI A full-service Kerberos environment consisting of a Kerberos server, a number of clients, and a number of application servers requires the following:

1. The Kerberos server must have the user ID and hashed passwords of all partic- ipating users in its database. All users are registered with the Kerberos server.

2. The Kerberos server must share a secret key with each server. All servers are registered with the Kerberos server.

15.3 / KERBEROS 491

Message (1) Client requests ticket-granting ticket. IDC Tells AS identity of user from this client. IDtgs Tells AS that user requests access to TGS. TS1 Allows AS to verify that client’s clock is synchronized with that of AS.

Message (2) AS returns ticket-granting ticket. Kc Encryption is based on user’s password, enabling AS and client to verify password, and

protecting contents of message (2). Kc, tgs Copy of session key accessible to client created by AS to permit secure exchange between

client and TGS without requiring them to share a permanent key. IDtgs Confirms that this ticket is for the TGS. TS2 Informs client of time this ticket was issued. Lifetime2 Informs client of the lifetime of this ticket.

Tickettgs Ticket to be used by client to access TGS.

(a) Authentication Service Exchange

Message (3) Client requests service-granting ticket. IDV Tells TGS that user requests access to server V. Tickettgs Assures TGS that this user has been authenticated by AS.

Authenticatorc Generated by client to validate ticket.

Table 15.2 Rationale for the Elements of the Kerberos Version 4 Protocol

Figure 15.3 Kerberos Exchanges

Client

Client authentication IDc || IDtgs || TS1

Tickettgs, server ID, and client authentication IDv || Tickettgs || Authenticatorc

Shared key and ticket E(Kc,tgs, [Kc,v || IDv || TS4 || Ticketv])

Ticketv and client authentication Ticketv || Authenticatorc

Service granted E(Kc,v, [TS5 + 1])

Shared key and ticket E(Kc, [Kc, tgs || IDtgs || TS2 ||

Lifetime2 || Tickettgs])

Authentication server (AS)

Ticket-granting server (TGS)

Service provider

(Continued)

492 CHAPTER 15 / USER AUTHENTICATION

Message (4) TGS returns service-granting ticket. Kc, tgs Key shared only by C and TGS protects contents of message (4). Kc, v Copy of session key accessible to client created by TGS to permit secure exchange between

client and server without requiring them to share a permanent key. IDV Confirms that this ticket is for server V. TS4 Informs client of time this ticket was issued. TicketV Ticket to be used by client to access server V.

Tickettgs Reusable so that user does not have to reenter password. Ktgs Ticket is encrypted with key known only to AS and TGS, to prevent tampering. Kc, tgs Copy of session key accessible to TGS used to decrypt authenticator, thereby authenticating

ticket. IDC Indicates the rightful owner of this ticket. ADC Prevents use of ticket from workstation other than one that initially requested the ticket. IDtgs Assures server that it has decrypted ticket properly. TS2 Informs TGS of time this ticket was issued. Lifetime2 Prevents replay after ticket has expired.

Authenticatorc Assures TGS that the ticket presenter is the same as the client for whom the ticket was issued has very short lifetime to prevent replay.

Kc, tgs Authenticator is encrypted with key known only to client and TGS, to prevent tampering. IDC Must match ID in ticket to authenticate ticket. ADC Must match address in ticket to authenticate ticket.

TS3 Informs TGS of time this authenticator was generated.

(b) Ticket-Granting Service Exchange

Message (5) Client requests service. TicketV Assures server that this user has been authenticated by AS.

Authenticatorc Generated by client to validate ticket. Message (6) Optional authentication of server to client.

Kc, v Assures C that this message is from V. TS5 + 1 Assures C that this is not a replay of an old reply. Ticketv Reusable so that client does not need to request a new ticket from TGS for each access to

the same server. Kv Ticket is encrypted with key known only to TGS and server, to prevent tampering. Kc, v Copy of session key accessible to client; used to decrypt authenticator, thereby authenticating

ticket. IDC Indicates the rightful owner of this ticket. ADC Prevents use of ticket from workstation other than one that initially requested the ticket. IDV Assures server that it has decrypted ticket properly. TS4 Informs server of time this ticket was issued. Lifetime4 Prevents replay after ticket has expired.

Authenticatorc Assures server that the ticket presenter is the same as the client for whom the ticket was issued; has very short lifetime to prevent replay.

Kc, v Authenticator is encrypted with key known only to client and server, to prevent tampering. IDC Must match ID in ticket to authenticate ticket. ADC Must match address in ticket to authenticate ticket.

TS5 Informs server of time this authenticator was generated.

(c) Client/Server Authentication Exchange

Table 15.2 Continued

15.3 / KERBEROS 493

Such an environment is referred to as a Kerberos realm. The concept of realm can be explained as follows. A Kerberos realm is a set of managed nodes that share the same Kerberos database. The Kerberos database resides on the Kerberos master computer system, which should be kept in a physically secure room. A read-only copy of the Kerberos database might also reside on other Kerberos computer systems. However, all changes to the database must be made on the master computer system. Changing or accessing the contents of a Kerberos database requires the Kerberos master password. A related concept is that of a Kerberos principal, which is a service or user that is known to the Kerberos system. Each Kerberos principal is identified by its principal name. Principal names consist of three parts: a service or user name, an instance name, and a realm name.

Networks of clients and servers under different administrative organizations typically constitute different realms. That is, it generally is not practical or does not conform to administrative policy to have users and servers in one administra- tive domain registered with a Kerberos server elsewhere. However, users in one realm may need access to servers in other realms, and some servers may be will- ing to provide service to users from other realms, provided that those users are authenticated.

Kerberos provides a mechanism for supporting such interrealm authentication. For two realms to support interrealm authentication, a third requirement is added:

3. The Kerberos server in each interoperating realm shares a secret key with the server in the other realm. The two Kerberos servers are registered with each other.

The scheme requires that the Kerberos server in one realm trust the Kerberos server in the other realm to authenticate its users. Furthermore, the participating servers in the second realm must also be willing to trust the Kerberos server in the first realm.

With these ground rules in place, we can describe the mechanism as follows (Figure 15.4): A user wishing service on a server in another realm needs a ticket for that server. The user’s client follows the usual procedures to gain access to the local TGS and then requests a ticket-granting ticket for a remote TGS (TGS in another realm). The client can then apply to the remote TGS for a service-granting ticket for the desired server in the realm of the remote TGS.

The details of the exchanges illustrated in Figure 15.4 are as follows (compare Table 15.1).

(1) C S AS: IDc } IDtgs }TS1 (2) AS S C: E(Kc, [Kc, tgs } IDtgs }TS2 }Lifetime2 }Tickettgs])

(3) C S TGS: IDtgsrem }Tickettgs }Authenticatorc (4) TGS S C: E(Kc,tgs, [Kc, tgsrem } IDtgsrem }TS4 }Tickettgsrem])

(5) C S TGSrem: IDvrem }Tickettgsrem }Authenticatorc (6) TGSrem S C: E(Kc,tgsrem, [Kc, vrem } IDvrem }TS6 }Ticketvrem])

(7) C S Vrem: Ticketvrem }Authenticatorc

494 CHAPTER 15 / USER AUTHENTICATION

The ticket presented to the remote server (Vrem) indicates the realm in which the user was originally authenticated. The server chooses whether to honor the re- mote request.

One problem presented by the foregoing approach is that it does not scale well to many realms. If there are N realms, then there must be N(N – 1)/2 secure key exchanges so that each Kerberos realm can interoperate with all other Kerberos realms.

Kerberos Version 5

Kerberos version 5 is specified in RFC 4120 and provides a number of improve- ments over version 4 [KOHL94]. To begin, we provide an overview of the changes from version 4 to version 5 and then look at the version 5 protocol.

Figure 15.4 Request for Service in Another Realm

Authentication server (AS)

Ticket- granting

server (TGS)

Kerberos

Authentication server (AS)

Ticket- granting

server (TGS)

Kerberos

Client

Realm A

Host/ application

server

Realm B

1. Requ est tick

et for l ocal TG

S

2. Tick et for l

ocal T GS

3. Request ticket for remoteTGS 4. Ticket for remote TGS

5. Request ticket

for rem ote server

6. Ticket for rem ote server

7. R

eq ue

st re

m ot

e se

rv ic

e

15.3 / KERBEROS 495

DIFFERENCES BETWEEN VERSIONS 4 AND 5 Version 5 is intended to address the limita- tions of version 4 in two areas: environmental shortcomings and technical deficien- cies. Let us briefly summarize the improvements in each area.8

Kerberos version 4 was developed for use within the Project Athena environ- ment and, accordingly, did not fully address the need to be of general purpose. This led to the following environmental shortcomings.

1. Encryption system dependence: Version 4 requires the use of DES. Export restriction on DES as well as doubts about the strength of DES were thus of concern. In version 5, ciphertext is tagged with an encryption-type identifier so that any encryption technique may be used. Encryption keys are tagged with a type and a length, allowing the same key to be used in different al- gorithms and allowing the specification of different variations on a given algorithm.

2. Internet protocol dependence: Version 4 requires the use of Internet Protocol (IP) addresses. Other address types, such as the ISO network address, are not accommodated. Version 5 network addresses are tagged with type and length, allowing any network address type to be used.

3. Message byte ordering: In version 4, the sender of a message employs a byte ordering of its own choosing and tags the message to indicate least signifi- cant byte in lowest address or most significant byte in lowest address. This techniques works but does not follow established conventions. In version 5, all message structures are defined using Abstract Syntax Notation One (ASN.1) and Basic Encoding Rules (BER), which provide an unambiguous byte ordering.

4. Ticket lifetime: Lifetime values in version 4 are encoded in an 8-bit quantity in units of five minutes. Thus, the maximum lifetime that can be expressed is 28 * 5 = 1280 minutes (a little over 21 hours). This may be inadequate for some applications (e.g., a long-running simulation that requires valid Kerberos credentials throughout execution). In version 5, tickets include an explicit start time and end time, allowing tickets with arbitrary lifetimes.

5. Authentication forwarding: Version 4 does not allow credentials issued to one client to be forwarded to some other host and used by some other client. This capability would enable a client to access a server and have that server access another server on behalf of the client. For example, a client issues a request to a print server that then accesses the client’s file from a file server, using the cli- ent’s credentials for access. Version 5 provides this capability.

6. Interrealm authentication: In version 4, interoperability among N realms requires on the order of N2 Kerberos-to-Kerberos relationships, as described earlier. Version 5 supports a method that requires fewer relationships, as de- scribed shortly.

8The following discussion follows the presentation in [KOHL94].

496 CHAPTER 15 / USER AUTHENTICATION

Apart from these environmental limitations, there are technical deficiencies in the version 4 protocol itself. Most of these deficiencies were documented in [BELL90], and version 5 attempts to address these. The deficiencies are the following.

1. Double encryption: Note in Table 15.1 [messages (2) and (4)] that tickets pro- vided to clients are encrypted twice—once with the secret key of the target server and then again with a secret key known to the client. The second en- cryption is not necessary and is computationally wasteful.

2. PCBC encryption: Encryption in version 4 makes use of a nonstandard mode of DES known as propagating cipher block chaining (PCBC).9 It has been demonstrated that this mode is vulnerable to an attack involving the inter- change of ciphertext blocks [KOHL89]. PCBC was intended to provide an in- tegrity check as part of the encryption operation. Version 5 provides explicit integrity mechanisms, allowing the standard CBC mode to be used for encryp- tion. In particular, a checksum or hash code is attached to the message prior to encryption using CBC.

3. Session keys: Each ticket includes a session key that is used by the client to encrypt the authenticator sent to the service associated with that ticket. In addition, the session key may subsequently be used by the client and the server to protect messages passed during that session. However, because the same ticket may be used repeatedly to gain service from a particular server, there is the risk that an opponent will replay messages from an old session to the client or the server. In version 5, it is possible for a client and server to negotiate a subsession key, which is to be used only for that one connection. A new access by the client would result in the use of a new subsession key.

4. Password attacks: Both versions are vulnerable to a password attack. The mes- sage from the AS to the client includes material encrypted with a key based on the client’s password.10 An opponent can capture this message and attempt to decrypt it by trying various passwords. If the result of a test decryption is of the proper form, then the opponent has discovered the client’s password and may subsequently use it to gain authentication credentials from Kerberos. This is the same type of password attack described in Chapter 21, with the same kinds of countermeasures being applicable. Version 5 does provide a mecha- nism known as preauthentication, which should make password attacks more difficult, but it does not prevent them.

THE VERSION 5 AUTHENTICATION DIALOGUE Table 15.3 summarizes the basic ver- sion 5 dialogue. This is best explained by comparison with version 4 (Table 15.1).

First, consider the authentication service exchange. Message (1) is a client re- quest for a ticket-granting ticket. As before, it includes the ID of the user and the TGS. The following new elements are added:

9This is described in Appendix T. 10Appendix T describes the mapping of passwords to encryption keys.

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15.3 / KERBEROS 497

(1) C S AS Options } IDc }Realmc } IDtgs }Times }Nonce1 (2) AS S C RealmC } IDC }Tickettgs }E(Kc, [Kc,tgs }Times }Nonce1 }Realmtgs } IDtgs])

Tickettgs = E(Ktgs, [Flags }Kc,tgs }Realmc } IDC }ADC }Times])

(a) Authentication Service Exchange to obtain ticket-granting ticket

(3) C S TGS Options } IDv }Times }Nonce2 }Tickettgs }Authenticatorc (4) TGS S C Realmc } IDC }Ticketv }E(Kc,tgs, [Kc,v }Times }Nonce2 }Realmv } IDv])

Tickettgs = E(Ktgs, [Flags }Kc,tgs }Realmc } IDC }ADC }Times]) Ticketv = E(Kv, [Flags }Kc,v }Realmc } IDC }ADC }Times])

Authenticatorc = E(Kc,tgs, [IDC }Realmc }TS1])

(b) Ticket-Granting Service Exchange to obtain service-granting ticket

(5) C S V Options }Ticketv }Authenticatorc (6) V S C EKc,v[TS2 }Subkey }Seq #]

Ticketv = E(Kv, [Flag }Kc,v }Realmc } IDC }ADC }Times]) Authenticatorc = E(Kc,v, [IDC }Relamc }TS2 }Subkey }Seq #])

(c) Client/Server Authentication Exchange to obtain service

Table 15.3 Summary of Kerberos Version 5 Message Exchanges

■ Realm: Indicates realm of user

■ Options: Used to request that certain flags be set in the returned ticket

■ Times: Used by the client to request the following time settings in the ticket:

—from: the desired start time for the requested ticket —till: the requested expiration time for the requested ticket —rtime: requested renew-till time

■ Nonce: A random value to be repeated in message (2) to assure that the re- sponse is fresh and has not been replayed by an opponent

Message (2) returns a ticket-granting ticket, identifying information for the client, and a block encrypted using the encryption key based on the user’s password. This block includes the session key to be used between the client and the TGS, times specified in message (1), the nonce from message (1), and TGS identifying information. The ticket itself includes the session key, identifying information for the client, the requested time values, and flags that reflect the status of this ticket and the requested options. These flags introduce significant new functionality to version 5. For now, we defer a discussion of these flags and concentrate on the over- all structure of the version 5 protocol.

Let us now compare the ticket-granting service exchange for versions 4 and 5. We see that message (3) for both versions includes an authenticator, a ticket, and the name of the requested service. In addition, version 5 includes re- quested times and options for the ticket and a nonce—all with functions similar to those of message (1). The authenticator itself is essentially the same as the one used in version 4.

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Message (4) has the same structure as message (2). It returns a ticket plus information needed by the client, with the information encrypted using the session key now shared by the client and the TGS.

Finally, for the client/server authentication exchange, several new features appear in version 5. In message (5), the client may request as an option that mutual authentication is required. The authenticator includes several new fields:

■ Subkey: The client’s choice for an encryption key to be used to protect this specific application session. If this field is omitted, the session key from the ticket (Kc,v) is used.

■ Sequence number: An optional field that specifies the starting sequence num- ber to be used by the server for messages sent to the client during this session. Messages may be sequence numbered to detect replays.

If mutual authentication is required, the server responds with message (6). This message includes the timestamp from the authenticator. Note that in version 4, the timestamp was incremented by one. This is not necessary in version 5, because the nature of the format of messages is such that it is not possible for an oppo- nent to create message (6) without knowledge of the appropriate encryption keys. The subkey field, if present, overrides the subkey field, if present, in message (5). The optional sequence number field specifies the starting sequence number to be used by the client.

TICKET FLAGS The flags field included in tickets in version 5 supports expanded functionality compared to that available in version 4. Table 15.4 summarizes the flags that may be included in a ticket.

INITIAL This ticket was issued using the AS protocol and not issued based on a ticket-granting ticket.

PRE-AUTHENT During initial authentication, the client was authenticated by the KDC before a ticket was issued.

HW-AUTHENT The protocol employed for initial authentication required the use of hard- ware expected to be possessed solely by the named client.

RENEWABLE Tells TGS that this ticket can be used to obtain a replacement ticket that expires at a later date.

MAY-POSTDATE Tells TGS that a postdated ticket may be issued based on this ticket- granting ticket.

POSTDATED Indicates that this ticket has been postdated; the end server can check the authtime field to see when the original authentication occurred.

INVALID This ticket is invalid and must be validated by the KDC before use. PROXIABLE Tells TGS that a new service-granting ticket with a different network

address may be issued based on the presented ticket. PROXY Indicates that this ticket is a proxy. FORWARDABLE Tells TGS that a new ticket-granting ticket with a different network

address may be issued based on this ticket-granting ticket. FORWARDED Indicates that this ticket has either been forwarded or was issued based on

authentication involving a forwarded ticket-granting ticket.

Table 15.4 Kerberos Version 5 Flags

15.3 / KERBEROS 499

The INITIAL flag indicates that this ticket was issued by the AS, not by the TGS. When a client requests a service-granting ticket from the TGS, it presents a ticket-granting ticket obtained from the AS. In version 4, this was the only way to obtain a service-granting ticket. Version 5 provides the additional capability that the client can get a service-granting ticket directly from the AS. The utility of this is as follows: A server, such as a password-changing server, may wish to know that the client’s password was recently tested.

The PRE-AUTHENT flag, if set, indicates that when the AS received the ini- tial request [message (1)], it authenticated the client before issuing a ticket. The exact form of this preauthentication is left unspecified. As an example, the MIT implementation of version 5 has encrypted timestamp preauthentication, enabled by default. When a user wants to get a ticket, it has to send to the AS a preauthen- tication block containing a random confounder, a version number, and a timestamp all encrypted in the client’s password-based key. The AS decrypts the block and will not send a ticket-granting ticket back unless the timestamp in the preauthentica- tion block is within the allowable time skew (time interval to account for clock drift and network delays). Another possibility is the use of a smart card that generates continually changing passwords that are included in the preauthenticated messages. The passwords generated by the card can be based on a user’s password but be transformed by the card so that, in effect, arbitrary passwords are used. This pre- vents an attack based on easily guessed passwords. If a smart card or similar device was used, this is indicated by the HW-AUTHENT flag.

When a ticket has a long lifetime, there is the potential for it to be stolen and used by an opponent for a considerable period. If a short lifetime is used to lessen the threat, then overhead is involved in acquiring new tickets. In the case of a ticket- granting ticket, the client would either have to store the user’s secret key, which is clearly risky, or repeatedly ask the user for a password. A compromise scheme is the use of renewable tickets. A ticket with the RENEWABLE flag set includes two expiration times: One for this specific ticket and one that is the latest permissible value for an expiration time. A client can have the ticket renewed by presenting it to the TGS with a requested new expiration time. If the new time is within the limit of the latest permissible value, the TGS can issue a new ticket with a new session time and a later specific expiration time. The advantage of this mechanism is that the TGS may refuse to renew a ticket reported as stolen.

A client may request that the AS provide a ticket-granting ticket with the MAY-POSTDATE flag set. The client can then use this ticket to request a ticket that is flagged as POSTDATED and INVALID from the TGS. Subsequently, the client may submit the postdated ticket for validation. This scheme can be useful for running a long batch job on a server that requires a ticket periodically. The client can obtain a number of tickets for this session at once, with spread out time values. All but the first ticket are initially invalid. When the execution reaches a point in time when a new ticket is required, the client can get the appropriate ticket validated. With this approach, the client does not have to repeatedly use its ticket- granting ticket to obtain a service-granting ticket.

In version 5, it is possible for a server to act as a proxy on behalf of a client, in effect adopting the credentials and privileges of the client to request a service from another server. If a client wishes to use this mechanism, it requests a ticket-granting

500 CHAPTER 15 / USER AUTHENTICATION

ticket with the PROXIABLE flag set. When this ticket is presented to the TGS, the TGS is permitted to issue a service-granting ticket with a different network address; this latter ticket will have its PROXY flag set. An application receiving such a ticket may accept it or require additional authentication to provide an audit trail.11

The proxy concept is a limited case of the more powerful forwarding procedure. If a ticket is set with the FORWARDABLE flag, a TGS can issue to the requestor a ticket-granting ticket with a different network address and the FORWARDED flag set. This ticket then can be presented to a remote TGS. This capability allows a cli- ent to gain access to a server on another realm without requiring that each Kerberos maintain a secret key with Kerberos servers in every other realm. For example, realms could be structured hierarchically. Then a client could walk up the tree to a common node and then back down to reach a target realm. Each step of the walk would involve forwarding a ticket-granting ticket to the next TGS in the path.

15.4 REMOTE USER-AUTHENTICATION USING ASYMMETRIC ENCRYPTION

Mutual Authentication

In Chapter 14, we presented one approach to the use of public-key encryption for the purpose of session-key distribution (Figure 14.9). This protocol assumes that each of the two parties is in possession of the current public key of the other. It may not be practical to require this assumption.

A protocol using timestamps is provided in [DENN81]:

1. A S AS: IDA } IDB 2. AS S A: E(PRas, [IDA }PUa }T]) }E(PRas, [IDB }PUb }T]) 3. A S B: E(PRas, [IDA }PUa }T]) }E(PRas, [IDB }PUb }T]) }

E(PUb, E(PRa, [Ks }T]))

In this case, the central system is referred to as an authentication server (AS), because it is not actually responsible for secret-key distribution. Rather, the AS pro- vides public-key certificates. The session key is chosen and encrypted by A; hence, there is no risk of exposure by the AS. The timestamps protect against replays of compromised keys.

This protocol is compact but, as before, requires the synchronization of clocks. Another approach, proposed by Woo and Lam [WOO92a], makes use of nonces. The protocol consists of the following steps.

1. A S KDC: IDA } IDB 2. KDC S A: E(PRauth, [IDB }PUb]) 3. A S B: E(PUb, [Na } IDA]) 4. B S KDC: IDA } IDB }E(PUauth, Na) 5. KDC S B: E(PRauth, [IDA }PUa]) }E(PUb, E(PRauth, [Na }Ks } IDB]))

11For a discussion of some of the possible uses of the proxy capability, see [NEUM93b].

15.4 / REMOTE USER-AUTHENTICATION USING ASYMMETRIC ENCRYPTION 501

6. B S A: E(PUa, [E(PRauth, [(Na }Ks } IDB)]) }Nb]) 7. A S B: E(Ks, Nb)

In step 1, A informs the KDC of its intention to establish a secure connection with B. The KDC returns to A a copy of B’s public-key certificate (step 2). Using B’s public key, A informs B of its desire to communicate and sends a nonce Na (step 3). In step 4, B asks the KDC for A’s public-key certificate and requests a session key; B includes A’s nonce so that the KDC can stamp the session key with that nonce. The nonce is protected using the KDC’s public key. In step 5, the KDC returns to B a copy of A’s public-key certificate, plus the information {Na, Ks, IDB}. This infor- mation basically says that Ks is a secret key generated by the KDC on behalf of B and tied to Na; the binding of Ks and Na will assure A that Ks is fresh. This triple is encrypted using the KDC’s private key to allow B to verify that the triple is in fact from the KDC. It is also encrypted using B’s public key so that no other entity may use the triple in an attempt to establish a fraudulent connection with A. In step 6, the triple {Na, Ks, IDB}, still encrypted with the KDC’s private key, is relayed to A, together with a nonce Nb generated by B. All the foregoing are encrypted using A’s public key. A retrieves the session key Ks, uses it to encrypt Nb, and returns it to B. This last message assures B of A’s knowledge of the session key.

This seems to be a secure protocol that takes into account the various attacks. However, the authors themselves spotted a flaw and submitted a revised version of the algorithm in [WOO92b]:

1. A S KDC: IDA } IDB 2. KDC S A: E(PRauth, [IDB }PUb]) 3. A S B: E(PUb, [Na } IDA]) 4. B S KDC: IDA } IDB }E(PUauth, Na) 5. KDC S B: E(PRauth, [IDA }PUa]) }E(PUb, E(PRauth, [Na }Ks } IDA } IDB])) 6. B S A: E(PUa, [Nb }E(PRauth, [Na }Ks } IDA } IDB])]) 7. A S B: E(Ks, Nb)

The identifier of A, IDA, is added to the set of items encrypted with the KDC’s private key in steps 5 and 6. This binds the session key Ks to the identities of the two parties that will be engaged in the session. This inclusion of IDA accounts for the fact that the nonce value Na is considered unique only among all nonces generated by A, not among all nonces generated by all parties. Thus, it is the pair {IDA, Na} that uniquely identifies the connection request of A.

In both this example and the protocols described earlier, protocols that ap- peared secure were revised after additional analysis. These examples highlight the difficulty of getting things right in the area of authentication.

One-Way Authentication

We have already presented public-key encryption approaches that are suited to electronic mail, including the straightforward encryption of the entire message for confidentiality (Figure 12.1b), authentication (Figure 12.1c), or both (Figure 12.1d). These approaches require that either the sender know the recipient’s public key

502 CHAPTER 15 / USER AUTHENTICATION

(confidentiality), the recipient know the sender’s public key (authentication), or both (confidentiality plus authentication). In addition, the public-key algorithm must be applied once or twice to what may be a long message.

If confidentiality is the primary concern, then the following may be more efficient:

A S B: E(PUb, Ks) }E(Ks, M)

In this case, the message is encrypted with a one-time secret key. A also encrypts this one-time key with B’s public key. Only B will be able to use the corresponding private key to recover the one-time key and then use that key to decrypt the message. This scheme is more efficient than simply encrypting the entire message with B’s public key.

If authentication is the primary concern, then a digital signature may suffice, as was illustrated in Figure 13.2:

A S B: M }E(PRa, H(M))

This method guarantees that A cannot later deny having sent the message. However, this technique is open to another kind of fraud. Bob composes a mes- sage to his boss Alice that contains an idea that will save the company money. He appends his digital signature and sends it into the email system. Eventually, the message will get delivered to Alice’s mailbox. But suppose that Max has heard of Bob’s idea and gains access to the mail queue before delivery. He finds Bob’s mes- sage, strips off his signature, appends his, and requeues the message to be delivered to Alice. Max gets credit for Bob’s idea.

To counter such a scheme, both the message and signature can be encrypted with the recipient’s public key:

A S B: E(PUb, [M }E(PRa, H(M))])

The latter two schemes require that B know A’s public key and be convinced that it is timely. An effective way to provide this assurance is the digital certificate, described in Chapter 14. Now we have

A S B: M }E(PRa, H(M)) }E(PRas, [T } IDA }PUa])

In addition to the message, A sends B the signature encrypted with A’s private key and A’s certificate encrypted with the private key of the authentication server. The recipient of the message first uses the certificate to obtain the sender’s public key and verify that it is authentic and then uses the public key to verify the message itself. If confidentiality is required, then the entire message can be encrypted with B’s public key. Alternatively, the entire message can be encrypted with a one-time secret key; the secret key is also transmitted, encrypted with B’s public key. This ap- proach is explored in Chapter 19.

15.5 FEDERATED IDENTITY MANAGEMENT

Federated identity management is a relatively new concept dealing with the use of a common identity management scheme across multiple enterprises and numerous applications and supporting many thousands, even millions, of users. We begin our overview with a discussion of the concept of identity management and then examine federated identity management.

15.5 / FEDERATED IDENTITY MANAGEMENT 503

Identity Management

Identity management is a centralized, automated approach to provide enterprise- wide access to resources by employees and other authorized individuals. The focus of identity management is defining an identity for each user (human or process), associating attributes with the identity, and enforcing a means by which a user can verify identity. The central concept of an identity management system is the use of single sign-on (SSO).

SSO enables a user to access all network resources after a single authentication. Typical services provided by a federated identity management system include

the following:

■ Point of contact: Includes authentication that a user corresponds to the user name provided, and management of user/server sessions.

■ SSO protocol services: Provides a vendor-neutral security token service for supporting a single sign on to federated services.

■ Trust services: Federation relationships require a trust relationship-based federation between business partners. A trust relationship is represented by the combination of the security tokens used to exchange information about a user, the cryptographic information used to protect these security tokens, and optionally the identity mapping rules applied to the information contained within this token.

■ Key services: Management of keys and certificates.

■ Identity services: services that provide the interface to local data stores, includ- ing user registries and databases, for identity-related information management.

■ Authorization: Granting access to specific services and/or resources based on the authentication.

■ Provisioning: Includes creating an account in each target system for the user, enrollment or registration of user in accounts, establishment of access rights or credentials to ensure the privacy and integrity of account data.

■ Management: Services related to runtime configuration and deployment.

Note that Kerberos contains a number of the elements of an identity manage- ment system.

Figure 15.5 illustrates entities and data flows in a generic identity manage- ment architecture. A principal is an identity holder. Typically, this is a human user that seeks access to resources and services on the network. User devices, agent pro- cesses, and server systems may also function as principals. Principals authenticate themselves to an identity provider. The identity provider associates authentication information with a principal, as well as attributes and one or more identifiers.

Increasingly, digital identities incorporate attributes other than simply an iden- tifier and authentication information (such as passwords and biometric information). An attribute service manages the creation and maintenance of such attributes. For example, a user needs to provide a shipping address each time an order is placed at a new Web merchant, and this information needs to be revised when the user moves. Identity management enables the user to provide this information once, so that it is maintained in a single place and released to data consumers in accordance with

504 CHAPTER 15 / USER AUTHENTICATION

authorization and privacy policies. Users may create some of the attributes to be associated with their digital identity, such as an address. Administrators may also as- sign attributes to users, such as roles, access permissions, and employee information.

Data consumers are entities that obtain and employ data maintained and provided by identity and attribute providers, which are often used to support autho- rization decisions and to collect audit information. For example, a database server or file server is a data consumer that needs a client’s credentials so as to know what access to provide to that client.

Identity Federation

Identity federation is, in essence, an extension of identity management to multiple security domains. Such domains include autonomous internal business units, exter- nal business partners, and other third-party applications and services. The goal is to provide the sharing of digital identities so that a user can be authenticated a single time and then access applications and resources across multiple domains. Because these domains are relatively autonomous or independent, no centralized control is possible. Rather, the cooperating organizations must form a federation based on agreed standards and mutual levels of trust to securely share digital identities.

Federated identity management refers to the agreements, standards, and technologies that enable the portability of identities, identity attributes, and entitle- ments across multiple enterprises and numerous applications and supporting many thousands, even millions, of users. When multiple organizations implement interop- erable federated identity schemes, an employee in one organization can use a single sign-on to access services across the federation with trust relationships associated with the identity. For example, an employee may log onto her corporate intranet and be authenticated to perform authorized functions and access authorized ser- vices on that intranet. The employee could then access their health benefits from an outside health-care provider without having to reauthenticate.

Figure 15.5 Generic Identity Management Architecture

Identity provider

Attribute service

Data consumer Principal

Administrator

15.5 / FEDERATED IDENTITY MANAGEMENT 505

Beyond SSO, federated identity management provides other capabilities. One is a standardized means of representing attributes. Increasingly, digital identities incorporate attributes other than simply an identifier and authentication informa- tion (such as passwords and biometric information). Examples of attributes include account numbers, organizational roles, physical location, and file ownership. A user may have multiple identifiers; for example, each identifier may be associated with a unique role with its own access permissions.

Another key function of federated identity management is identity mapping. Different security domains may represent identities and attributes differently. Further, the amount of information associated with an individual in one domain may be more than is necessary in another domain. The federated identity manage- ment protocols map identities and attributes of a user in one domain to the require- ments of another domain.

Figure 15.6 illustrates entities and data flows in a generic federated identity management architecture.

Figure 15.6 Federated Identity Operation

User

1 Identity provider (source domain)

Service provider (destination domain)

1 End user’s browser or other application engages in an authentication dialogue with identity provider in the same domain. End user also provides attribute values associated with user’s identity.

2 Some attributes associated with an identity, such as allowable roles, may be provided by an administrator in the same domain.

3 A service provider in a remote domain, which the user wishes to access, obtains identity information, authentication information, and associated attributes from the identity provider in the source domain.

4 Service provider opens session with remote user and enforces access control restrictions based on user’s identity and attributes.

Administrator

2

3

4

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The identity provider acquires attribute information through dialogue and pro- tocol exchanges with users and administrators. For example, a user needs to provide a shipping address each time an order is placed at a new Web merchant, and this information needs to be revised when the user moves. Identity management enables the user to provide this information once, so that it is maintained in a single place and released to data consumers in accordance with authorization and privacy policies.

Service providers are entities that obtain and employ data maintained and pro- vided by identity providers, often to support authorization decisions and to collect audit information. For example, a database server or file server is a data consumer that needs a client’s credentials so as to know what access to provide to that client. A service provider can be in the same domain as the user and the identity provider. The power of this approach is for federated identity management, in which the ser- vice provider is in a different domain (e.g., a vendor or supplier network).

STANDARDS Federated identity management uses a number of standards as the building blocks for secure identity exchange across different domains or heteroge- neous systems. In essence, organizations issue some form of security tickets for their users that can be processed by cooperating partners. Identity federation standards are thus concerned with defining these tickets, in terms of content and format, pro- viding protocols for exchanging tickets and performing a number of management tasks. These tasks include configuring systems to perform attribute transfers and identity mapping, and performing logging and auditing functions. The key stan- dards are as follows:

■ The Extensible Markup Language (XML): A markup language that uses sets of embedded tags or labels to characterize text elements within a document so as to indicate their appearance, function, meaning, or context. XML docu- ments appear similar to HTML (Hypertext Markup Language) documents that are visible as Web pages, but provide greater functionality. XML includes strict definitions of the data type of each field, thus supporting database for- mats and semantics. XML provides encoding rules for commands that are used to transfer and update data objects.

■ The Simple Object Access Protocol (SOAP): A minimal set of conventions for invoking code using XML over HTTP. It enables applications to request services from one another with XML-based requests and receive responses as data formatted with XML. Thus, XML defines data objects and structures, and SOAP provides a means of exchanging such data objects and performing remote procedure calls related to these objects. See [ROS06] for an informa- tive discussion.

■ WS-Security: A set of SOAP extensions for implementing message integrity and confidentiality in Web services. To provide for secure exchange of SOAP messages among applications, WS-Security assigns security tokens to each message for use in authentication.

■ Security Assertion Markup Language (SAML): An XML-based language for the exchange of security information between online business partners. SAML conveys authentication information in the form of assertions about subjects. Assertions are statements about the subject issued by an authoritative entity.

15.5 / FEDERATED IDENTITY MANAGEMENT 507

The challenge with federated identity management is to integrate multiple technologies, standards, and services to provide a secure, user-friendly utility. The key, as in most areas of security and networking, is the reliance on a few mature standards widely accepted by industry. Federated identity management seems to have reached this level of maturity.

EXAMPLES To get some feel for the functionality of identity federation, we look at three scenarios, taken from [COMP06].

In the first scenario (Figure 15.7a), Workplace.com contracts with Health.com to provide employee health benefits. An employee uses a Web interface to sign on to Workplace.com and goes through an authentication procedure there. This enables the employee to access authorized services and resources at Workplace.com. When the employee clicks on a link to access health benefits, her browser is redirected to Health.com. At the same time, the Workplace.com software passes the user’s identi- fier to Health.com in a secure manner. The two organizations are part of a federation that cooperatively exchanges user identifiers. Health.com maintains user identities

Figure 15.7 Federated Identity Scenarios

User store

(a) Federation based on account linking

(c) Chained Web services

Workplace.com (employee portal)

Name Joe Jane Ravi

ID 1213 1410 1603

User store Name Joe Jane Ravi

ID 1213 1410 1603

Health.com

User store

(b) Federation based on roles

Name Joe Jane Ravi

ID 1213 1410 1603

Dept Eng Purch Purch

User store Role

Engineer Purchaser

Aut hen

tica tion Website access

End user (employee)

User ID

Workplace.com (procurement application)

PinSupplies.com (Purchasing Web

service)

Aut hen

tica tion

Pro cur

eme nt

req ues

t

End user (employee)

SOAP message

Eship.com (shipping Web

service)

SOAP message

Workplace.com (employee portal)

PartsSupplier.com

Aut hen

tica tion Website access

End user (employee)

Role

508 CHAPTER 15 / USER AUTHENTICATION

for every employee at Workplace.com and associates with each identity health-bene- fits information and access rights. In this example, the linkage between the two com- panies is based on account information and user participation is browser based.

Figure 15.7b shows a second type of browser-based scheme. PartsSupplier. com is a regular supplier of parts to Workplace.com. In this case, a role-based access-control (RBAC) scheme is used for access to information. An engineer of Workplace.com authenticates at the employee portal at Workplace.com and clicks on a link to access information at PartsSupplier.com. Because the user is authen- ticated in the role of an engineer, he is taken to the technical documentation and troubleshooting portion of PartsSupplier.com’s Web site without having to sign on. Similarly, an employee in a purchasing role signs on at Workplace.com and is au- thorized, in that role, to place purchases at PartsSupplier.com without having to authenticate to PartsSupplier.com. For this scenario, PartsSupplier.com does not have identity information for individual employees at Workplace.com. Rather, the linkage between the two federated partners is in terms of roles.

The scenario illustrated in Figure 15.7c can be referred to as document based rather than browser based. In this third example, Workplace.com has a purchasing agreement with PinSupplies.com, and PinSupplies.com has a business relationship with E-Ship.com. An employee of Workplace.com signs on and is authenticated to make purchases. The employee goes to a procurement application that provides a list of Workplace.com’s suppliers and the parts that can be ordered. The user clicks on the PinSupplies button and is presented with a purchase order Web page (HTML page). The employee fills out the form and clicks the submit button. The procure- ment application generates an XML/SOAP document that it inserts into the enve- lope body of an XML-based message. The procurement application then inserts the user’s credentials in the envelope header of the message, together with Workplace. com’s organizational identity. The procurement application posts the message to the PinSupplies.com’s purchasing Web service. This service authenticates the in- coming message and processes the request. The purchasing Web service then sends a SOAP message to its shipping partner to fulfill the order. The message includes a PinSupplies.com security token in the envelope header and the list of items to be shipped as well as the end user’s shipping information in the envelope body. The shipping Web service authenticates the request and processes the shipment order.

15.6 PERSONAL IDENTITY VERIFICATION

User authentication based on the possession of a smart card is becoming more wide- spread. A smart card has the appearance of a credit card, has an electronic inter- face, and may use a variety of authentication protocols.

A smart card contains within it an entire microprocessor, including processor, memory, and I/O ports. Some versions incorporate a special co-processing circuit for cryptographic operation to speed the task of encoding and decoding messages or generating digital signatures to validate the information transferred. In some cards, the I/O ports are directly accessible by a compatible reader by means of exposed electrical contacts. Other cards rely instead on an embedded antenna for wireless communication with the reader.

15.6 / PERSONAL IDENTITY VERIFICATION 509

A typical smart card includes three types of memory. Read-only memory (ROM) stores data that does not change during the card’s life, such as the card number and the cardholder’s name. Electrically erasable programmable ROM (EEPROM) holds application data and programs, such as the protocols that the card can execute. It also holds data that may vary with time. For example, in a tele- phone card, the EEPROM holds the talk time remaining. Random access memory (RAM) holds temporary data generated when applications are executed.

For the practical application of smart card authentication, a wide range of vendors must conform to standards that cover smart card protocols, authentication and access control formats and protocols, database entries, message formats, and so on. An important step in this direction is FIPS 201-2 (Personal Identity Verification [PIV] of Federal Employees and Contractors, June 2012). The standard defines a reliable, government-wide PIV system for use in applications such as access to fed- erally controlled facilities and information systems. The standard specifies a PIV system within which common identification credentials can be created and later used to verify a claimed identity. The standard also identifies Federal government- wide requirements for security levels that are dependent on risks to the facility or information being protected. The standard applies to private-sector contractors as well, and serves as a useful guideline for any organization.

PIV System Model

Figure 15.8 illustrates the major components of FIPS 201-2 compliant systems. The PIV front end defines the physical interface to a user who is requesting access to a facility, which could be either physical access to a protected physical area or logical access to an information system. The PIV front-end subsystem supports up to three- factor authentication; the number of factors used depends on the level of security required. The front end makes use of a smart card, known as a PIV card, which is a dual-interface contact and contactless card. The card holds a cardholder pho- tograph, X.509 certificates, cryptographic keys, biometric data, and a cardholder unique identifier (CHUID). Certain cardholder information may be read-protected and require a personal identification number (PIN) for read access by the card reader. The biometric reader, in the current version of the standard, is a fingerprint reader or an iris scanner.

The standard defines three assurance levels for verification of the card and the encoded data stored on the card, which in turn leads to verifying the authenticity of the person holding the credential. A level of some confidence corresponds to use of the card reader and PIN. A level of high confidence adds a biometric comparison of a fingerprint captured and encoded on the card during the card-issuing process and a fingerprint scanned at the physical access point. A very high confidence level requires that the process just described is completed at a control point attended by an official observer.

The other major component of the PIV system is the PIV card issuance and management subsystem. This subsystem includes the components responsible for identity proofing and registration, card and key issuance and management, and the various repositories and services (e.g., public key infrastructure [PKI] directory, certificate status servers) required as part of the verification infrastructure.

510 CHAPTER 15 / USER AUTHENTICATION

The PIV system interacts with a relying subsystem, which includes compo- nents responsible for determining a particular PIV cardholder’s access to a physical or logical resource. FIPS 201-2 standardizes data formats and protocols for interac- tion between the PIV system and the relying system.

Unlike the typical card number/facility code encoded on most access control cards, the FIPS 201 CHUID takes authentication to a new level, through the use of an expiration date (a required CHUID data field) and an optional CHUID digital signature. A digital signature can be checked to ensure that the CHUID recorded on the card was digitally signed by a trusted source and that the CHUID data have not been altered since the card was signed. The CHUID expiration date can be checked to verify that the card has not expired. This is independent from whatever expiration date is associated with cardholder privileges. Reading and verifying the CHUID alone provides only some assurance of identity because it authenticates the card data, not the cardholder. The PIN and biometric factors provide identity verification of the individual.

PIV Documentation

The PIV specification is quite complex, and NIST has issued a number of docu- ments that cover a broad range of PIV topics. These are as follows:

Figure 15.8 FIPS 201 PIV System Model

Identity profiling & registration

Card issuance & maintenance

Key management

PKI directory & certificate status

responder

Authorization data

Authorization data

Physical resource

Logical resource

PIV card issuance and management

Shapes

Relying

I&A Authorization

Physical Access Control

I&A = Identification and Authentication

Authorization

Direction of information flow

Processes

Components

Logical Access Control

Card reader /writer

PIN input device

Biometric reader

PIV card

PIV Front End

LEGEND

I&A

15.6 / PERSONAL IDENTITY VERIFICATION 511

■ FIPS 201-2—Personal Identity Verification (PIV) of Federal Employees and Contractors: Specifies the physical card characteristics, storage media, and data elements that make up the identity credentials resident on the PIV card.

■ SP 800-73-3—Interfaces for Personal Identity Verification: Specifies the in- terfaces and card architecture for storing and retrieving identity credentials from a smart card, and provides guidelines for the use of authentication mech- anisms and protocols.

■ SP 800-76-2—Biometric Data Specification for Personal Identity Verification: Describes technical acquisition and formatting specifications for the biometric credentials of the PIV system.

■ SP 800-78-3—Cryptographic Algorithms and Key Sizes for Personal Identity Verification: Identifies acceptable symmetric and asymmetric encryption algo- rithms, digital signature algorithms, and message digest algorithms, and speci- fies mechanisms to identify the algorithms associated with PIV keys or digital signatures.

■ SP 800-104—A Scheme for PIV Visual Card Topography: Provides additional recommendations on the PIV card color-coding for designating employee affiliation.

■ SP 800-116—A Recommendation for the Use of PIV Credentials in Physical Access Control Systems (PACS): Describes a risk-based approach for select- ing appropriate PIV authentication mechanisms to manage physical access to Federal government facilities and assets.

■ SP 800-79-1—Guidelines for the Accreditation of Personal Identity Verification Card Issuers: Provides guidelines for accrediting the reliability of issuers of PIV cards that collect, store, and disseminate personal identity credentials and issue smart cards.

■ SP 800-96—PIV Card to Reader Interoperability Guidelines: Provides re- quirements that facilitate interoperability between any card and any reader.

In addition there are other documents that deal with conformance testing and codes for identifiers.

PIV Credentials and Keys

The PIV card contains a number of mandatory and optional data elements that serve as identity credentials with varying levels of strength and assurance. These credentials are used singly or in sets to authenticate the holder of the PIV card to achieve the level of assurance required for a particular activity or transaction. The mandatory data elements are the following:

■ Personal Identification Number (PIN): Required to activate the card for privi- leged operation.

■ Cardholder Unique Identifier (CHUID): Includes the Federal Agency Smart Credential Number (FASC-N) and the Global Unique Identification Number (GUID), which uniquely identify the card and the cardholder.

512 CHAPTER 15 / USER AUTHENTICATION

■ PIV Authentication Key: Asymmetric key pair and corresponding certificate for user authentication.

■ Two fingerprint templates: For biometric authentication.

■ Electronic facial image: For biometric authentication.

■ Asymmetric Card Authentication Key: Asymmetric key pair and correspond- ing certificate used for card authentication.

Optional elements include the following:

■ Digital Signature Key: Asymmetric key pair and corresponding certificate that supports document signing and signing of data elements such as the CHUID.

■ Key Management Key: Asymmetric key pair and corresponding certificate supporting key establishment and transport.

■ Symmetric Card Authentication Key: For supporting physical access applications.

■ PIV Card Application Administration Key: Symmetric key associated with the card management system.

■ One or two iris images: For biometric authentication.

Table 15.5 lists the algorithm and key size requirements for PIV key types.

Authentication

Using the electronic credentials resident on a PIV card, the card supports the fol- lowing authentication mechanisms:

■ CHUID: The cardholder is authenticated using the signed CHUID data ele- ment on the card. The PIN is not required. This mechanism is useful in envi- ronments where a low level of assurance is acceptable and rapid contactless authentication is necessary.

PIV Key Type Algorithms Key Sizes (bits) Application

PIV Authentication Key

RSA 2048 Supports card and cardholder authentication for an interoperable environmentECDSA 256

Card Authentication Key

3TDEA 168 Supports card authentication for physical accessAES 128, 192, or 256

RSA 2048 Supports card authentication for an interoperable environmentECDSA 256

Digital Signature Key RSA 2048 or 3072 Supports document signing

and nonce signingECDSA 256 or 384

Key Management Key RSA 2048 Supports key establishment

and transportECDH 256 or 384

Table 15.5 PIV Algorithms and Key Sizes

15.6 / PERSONAL IDENTITY VERIFICATION 513

■ Card Authentication Key: The PIV card is authenticated using the Card Authentication Key in a challenge response protocol. The PIN is not required. This mechanism allows contact (via card reader) or contactless (via radio waves) authentication of the PIV card without the holder’s active participa- tion, and provides a low level of assurance.

■ BIO: The cardholder is authenticated by matching his or her fingerprint sample(s) to the signed biometric data element in an environment without a human attendant in view. The PIN is required to activate the card. This mecha- nism achieves a high level of assurance and requires the cardholder’s active participation is submitting the PIN as well as the biometric sample.

■ BIO-A: The cardholder is authenticated by matching his or her fingerprint sample(s) to the signed biometric data element in an environment with a human attendant in view. The PIN is required to activate the card. This mecha- nism achieves a very high level of assurance when coupled with full trust val- idation of the biometric template retrieved from the card, and requires the cardholder’s active participation is submitting the PIN as well as the biometric sample.

■ PKI: The cardholder is authenticated by demonstrating control of the PIV au- thentication private key in a challenge response protocol that can be validated using the PIV authentication certificate. The PIN is required to activate the card. This mechanism achieves a very high level of identity assurance and re- quires the cardholder’s knowledge of the PIN.

In each of the above use cases, except the symmetric Card Authentication Key use case, the source and the integrity of the corresponding PIV credential are vali- dated by verifying the digital signature on the credential, with the signature being provided by a trusted entity.

A variety of protocols can be constructed for each of these authentication types. SP 800-78-3 gives examples for each type. Figure 15.9 illustrates an authenti- cation scenario that includes the use of the PIV Authentication Key. This scenario provides a high level of assurance. This scenario would be appropriate for authenti- cation of a user who possesses a PIV card and seeks access to a computer resource. The computer, designated local system in the figure, includes PIV application soft- ware and communicates to the card via an application program interface that en- ables the use of relatively high-level procedure calls. These high-level commands are converted into PIV commands that are issued to the card through a physical interface via a card reader or via a wireless interface. In either case, SP 800-73 refers to the card command interface as the PIV card edge.

The process begins when the local system detects the card either through an attached card reader or wirelessly. It then selects an application on the card for au- thentication. The local system then requests the public-key certificate for the card’s PIV Authentication Key. If the certificate is valid (i.e., has a valid signature, has not expired or been revoked), authentication continues. Otherwise the card is rejected. The next step is for the local system to request that the cardholder enter the PIN for the card. If the submitted PIN matches the PIN stored on the card, the card returns a positive acknowledgment; otherwise the card returns a failure message.

514 CHAPTER 15 / USER AUTHENTICATION

The local system either continues or rejects the card accordingly. The next phase is a challenge-response protocol. The local system sends a nonce to be signed by the PIV, and the PIV returns the signature. The local system uses the PIV authentica- tion public key to verify the signature. If the signature is valid, the cardholder is ac- cepted as being identified. Otherwise the local system rejects the card.

The scenario of Figure 15.9 accomplishes three types of authentication. The combination of possession of the card and knowledge of the PIN service authenti- cates the cardholder. The PIV Authentication Key certificate validates the card’s credentials. The challenge-response protocol authenticates the card.

Figure 15.9 Authentication Using PIV Authentication Key

Connect

Disconnect End transaction

Verify PIN

Request card signature

PIV card app ID and Version

PIV Auth certificate returned

Signed nonce returned

PIN ACK

Read value (PIV Auth certificate)

Select application Select application

Verify PIN

Sign nonce

Begin transaction

Present card (HolderV)

PIV Application on Local System

API on Local System

PIV Card Edge

Retrieve PIV AUTH certificate

Retrieve FASC-N from the certificate

CardV = Card validation CredV = Credential validation HolderV = Cardholder validation FASC-N = Federal Agency Smart Credential Number

Validate certificate (signature, expiration, and

revocation) (CredV)

Retrieve algorithm ID and key size for signature request

Acquire PIN (HolderV)

Verify signed data—card

possesses private key (CardV)

Reject Cardholder identifier

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15.7 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 515

15.7 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS

Key Terms

authentication authentication server claimant credential credential service provider

(CSP) federated identity

management identity management

Kerberos Kerberos realm mutual authentication nonce one-way authentication personal identity verification

(PIV) realm registration authority (RA)

relying party (RP) replay attack subscriber suppress-replay attack ticket ticket-granting server (TGS) timestamp verifier

Review Questions

15.1 What are the steps involved in an authentication process? 15.2 List three general approaches to dealing with replay attacks. 15.3 What is a suppress-replay attack? 15.4 What problem was Kerberos designed to address? 15.5 What are three threats associated with user authentication over a network or

Internet? 15.6 List three approaches to secure user authentication in a distributed environment. 15.7 What four requirements were defined for Kerberos? 15.8 What entities constitute a full-service Kerberos environment? 15.9 In the context of Kerberos, what is a realm? 15.10 What are the mandatory elements to authenticate a PIV card holder?

Problems

15.1 In Section 15.4, we outlined the public-key scheme proposed in [WOO92a] for the distribution of secret keys. The revised version includes IDA in steps 5 and 6. What attack, specifically, is countered by this revision?

15.2 The protocol referred to in Problem 15.1 can be reduced from seven steps to five, having the following sequence: a. A S B: b. A S KDC: c. KDC S B: d. B S A: e. A S B: Show the message transmitted at each step. Hint: The final message in this protocol is the same as the final message in the original protocol.

15.3 Reference the suppress-replay attack described in Section 15.2 to answer the following. a. Give an example of an attack when a party’s clock is ahead of that of the KDC. b. Give an example of an attack when a party’s clock is ahead of that of another

party.

516 CHAPTER 15 / USER AUTHENTICATION

15.4 There are three typical ways to use nonces as challenges. Suppose Na is a nonce gen- erated by A, A and B share key K, and f() is a function (such as an increment). The three usages are

Usage 1 Usage 2 Usage 3

(1) A S B: Na (1) A S B: E(K, Na) (1) A S B: E(K, Na) (2) B S A: E(K, Na) (2) B S A: Na (2) B S A: E(K, f(Na))

Describe situations for which each usage is appropriate. 15.5 Show that a random error in one block of ciphertext is propagated to all subsequent

blocks of plaintext in PCBC mode (See Figure T.2 in Appendix T). 15.6 Suppose that, in PCBC mode, blocks Ci and Ci+ 1 are interchanged during transmis-

sion. Show that this affects only the decrypted blocks Pi and Pi+ 1 but not subsequent blocks.

15.7 In addition to providing a standard for public-key certificate formats, X.509 specifies an authentication protocol. The original version of X.509 contains a security flaw. The essence of the protocol is as follows.

A S B: A {tA, rA, IDB} B S A: B {tB, rB, IDA, rA} A S B: A {rB}

where tA and tB are timestamps, rA and rB are nonces and the notation X{Y} indicates that the message Y is transmitted, encrypted, and signed by X.

The text of X.509 states that checking timestamps tA and tB is optional for three-way authentication. But consider the following example: Suppose A and B have used the preceding protocol on some previous occasion, and that opponent C has intercepted the preceding three messages. In addition, suppose that timestamps are not used and are all set to 0. Finally, suppose C wishes to impersonate A to B. C initially sends the first captured message to B:

C S B: A {0, rA, IDB}

B responds, thinking it is talking to A but is actually talking to C:

B S C: B {0, r B= , IDA, rA}

C meanwhile causes A to initiate authentication with C by some means. As a result, A sends C the following:

A S C: A {0, r A= , IDC}

C responds to A using the same nonce provided to C by B:

C S A: C {0, r B= , IDA, r A= }

A responds with

A S C: A {r B= }

15.7 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 517

This is exactly what C needs to convince B that it is talking to A, so C now repeats the incoming message back out to B.

C S B: A {r B= }

So B will believe it is talking to A whereas it is actually talking to C. Suggest a simple solution to this problem that does not involve the use of timestamps.

15.8 Consider a one-way authentication technique based on asymmetric encryption:

A S B: IDA B S A: R1 A S B: E(PRa, R1)

a. Explain the protocol. b. What type of attack is this protocol susceptible to?

15.9 Consider a one-way authentication technique based on asymmetric encryption:

A S B: IDA| | E(PUB,RA)

BS A: RA

a. Explain the protocol. b. What type of attack is this protocol susceptible to?

15.10 In Kerberos, when Bob receives a Ticket from Alice, how does he know it is not genuine?

15.11 In Kerberos, how does Bob know that the received token is not corresponding to Alice’s?

15.12 In Kerberos, how does Alice know that a reply to an earlier message is from Bob? 15.13 In Kerberos, what does the Ticket contain that allows Alice and Bob to talk securely?

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519

16.1 Network Access Control

Elements of a Network Access Control System Network Access Enforcement Methods

16.2 Extensible Authentication Protocol

Authentication Methods EAP Exchanges

16.3 IEEE 802.1X Port-Based Network Access Control

16.4 Cloud Computing

Cloud Computing Elements Cloud Computing Reference Architecture

16.5 Cloud Security Risks and Countermeasures

16.6 Data Protection in the Cloud

16.7 Cloud Security as a Service

16.8 Addressing Cloud Computing Security Concerns

16.9 Key Terms, Review Questions, and Problems

PART SIX: NETWORK AND INTERNET SECURITY

CHAPTER

Network Access Control and Cloud Security

520 CHAPTER 16 / NETWORK ACCESS CONTROL AND CLOUD SECURITY

This chapter begins our discussion of network security, focusing on two key topics: network access control and cloud security. We begin with an overview of network access control systems, summarizing the principal elements and techniques involved in such a system. Next, we discuss the Extensible Authentication Protocol and IEEE 802.1X, two widely implemented standards that are the foundation of many network access control systems.

The remainder of the chapter deals with cloud security. We begin with an overview of cloud computing, and follow this with a discussion of cloud security issues.

16.1 NETWORK ACCESS CONTROL

Network access control (NAC) is an umbrella term for managing access to a network. NAC authenticates users logging into the network and determines what data they can access and actions they can perform. NAC also examines the health of the user’s computer or mobile device (the endpoints).

Elements of a Network Access Control System

NAC systems deal with three categories of components:

■ Access requestor (AR): The AR is the node that is attempting to access the network and may be any device that is managed by the NAC system, including workstations, servers, printers, cameras, and other IP-enabled devices. ARs are also referred to as supplicants, or simply, clients.

■ Policy server: Based on the AR’s posture and an enterprise’s defined policy, the policy server determines what access should be granted. The policy server often relies on backend systems, including antivirus, patch management, or a user directory, to help determine the host’s condition.

LEARNING OBJECTIVES

After studying this chapter, you should be able to:

◆ Discuss the principal elements of a network access control system.

◆ Discuss the principal network access enforcement methods.

◆ Present an overview of the Extensible Authentication Protocol.

◆ Understand the operation and role of the IEEE 802.1X Port-Based Network Access Control mechanism.

◆ Present an overview of cloud computing concepts.

◆ Understand the unique security issues related to cloud computing.

16.1 / NETWORK ACCESS CONTROL 521

■ Network access server (NAS): The NAS functions as an access control point for users in remote locations connecting to an enterprise’s internal network. Also called a media gateway, a remote access server (RAS), or a policy server, an NAS may include its own authentication services or rely on a separate authentication service from the policy server.

Figure 16.1 is a generic network access diagram. A variety of different ARs seek access to an enterprise network by applying to some type of NAS. The first step is generally to authenticate the AR. Authentication typically involves some sort of secure protocol and the use of cryptographic keys. Authentication may be performed by the NAS, or the NAS may mediate the authentication process. In the latter case, authentication takes place between the supplicant and an authentication server that is part of the policy server or that is accessed by the policy server.

The authentication process serves a number of purposes. It verifies a suppli- cant’s claimed identity, which enables the policy server to determine what access privileges, if any, the AR may have. The authentication exchange may result in the

Figure 16.1 Network Access Control Context

Supplicants

Network access servers

Authentication server

DHCP server

VLAN server

Policy server

Patch management

Network resources

Quarantine network

Antivirus Antispyware

Enterprise network

522 CHAPTER 16 / NETWORK ACCESS CONTROL AND CLOUD SECURITY

establishment of session keys to enable future secure communication between the supplicant and resources on the enterprise network.

Typically, the policy server or a supporting server will perform checks on the AR to determine if it should be permitted interactive remote access connectivity. These checks—sometimes called health, suitability, screening, or assessment checks—require software on the user’s system to verify compliance with certain requirements from the organization’s secure configuration baseline. For example, the user’s antimalware software must be up-to-date, the operating system must be fully patched, and the remote computer must be owned and controlled by the organization. These checks should be performed before granting the AR access to the enterprise network. Based on the results of these checks, the organization can determine whether the remote computer should be permitted to use interactive remote access. If the user has acceptable authorization credentials but the remote computer does not pass the health check, the user and remote computer should be denied network access or have limited access to a quarantine network so that autho- rized personnel can fix the security deficiencies. Figure 16.1 indicates that the quar- antine portion of the enterprise network consists of the policy server and related AR suitability servers. There may also be application servers that do not require the normal security threshold be met.

Once an AR has been authenticated and cleared for a certain level of access to the enterprise network, the NAS can enable the AR to interact with resources in the enterprise network. The NAS may mediate every exchange to enforce a security policy for this AR, or may use other methods to limit the privileges of the AR.

Network Access Enforcement Methods

Enforcement methods are the actions that are applied to ARs to regulate access to the enterprise network. Many vendors support multiple enforcement methods simultaneously, allowing the customer to tailor the configuration by using one or a combination of methods. The following are common NAC enforcement methods.

■ IEEE 802.1X: This is a link layer protocol that enforces authorization before a port is assigned an IP address. IEEE 802.1X makes use of the Extensible Authentication Protocol for the authentication process. Sections 16.2 and 16.3 cover the Extensible Authentication Protocol and IEEE 802.1X, respectively.

■ Virtual local area networks (VLANs): In this approach, the enterprise net- work, consisting of an interconnected set of LANs, is segmented logically into a number of virtual LANs.1 The NAC system decides to which of the network’s VLANs it will direct an AR, based on whether the device needs security reme- diation, Internet access only, or some level of network access to enterprise resources. VLANs can be created dynamically and VLAN membership, of both enterprise servers and ARs, may overlap. That is, an enterprise server or an AR may belong to more than one VLAN.

1A VLAN is a logical subgroup within a LAN that is created via software rather than manually moving cables in the wiring closet. It combines user stations and network devices into a single unit regardless of the physical LAN segment they are attached to and allows traffic to flow more efficiently within populations of mutual interest. VLANs are implemented in port-switching hubs and LAN switches.

16.2 / EXTENSIBLE AUTHENTICATION PROTOCOL 523

■ Firewall: A firewall provides a form of NAC by allowing or denying network traffic between an enterprise host and an external user. Firewalls are discussed in Chapter 23.

■ DHCP management: The Dynamic Host Configuration Protocol (DHCP) is an Internet protocol that enables dynamic allocation of IP addresses to hosts. A DHCP server intercepts DHCP requests and assigns IP addresses instead. Thus, NAC enforcement occurs at the IP layer based on subnet and IP assign- ment. A DCHP server is easy to install and configure, but is subject to various forms of IP spoofing, providing limited security.

There are a number of other enforcement methods available from vendors. The ones in the preceding list are perhaps the most common, and IEEE 802.1X is by far the most commonly implemented solution.

16.2 EXTENSIBLE AUTHENTICATION PROTOCOL

The Extensible Authentication Protocol (EAP), defined in RFC 3748, acts as a framework for network access and authentication protocols. EAP provides a set of protocol messages that can encapsulate various authentication methods to be used between a client and an authentication server. EAP can operate over a variety of network and link level facilities, including point-to-point links, LANs, and other networks, and can accommodate the authentication needs of the various links and networks. Figure 16.2 illustrates the protocol layers that form the context for EAP.

Authentication Methods

EAP supports multiple authentication methods. This is what is meant by referring to EAP as extensible. EAP provides a generic transport service for the exchange of authentication information between a client system and an authentication server. The basic EAP transport service is extended by using a specific authentication proto- col, or method, that is installed in both the EAP client and the authentication server.

Figure 16.2 EAP Layered Context

Authentication methods

EAP layer

Data link layer

Extensible Authentication Protocol (EAP)

IEEE 802.1X EAP over LAN (EAPOL)

EAP- TLS

EAP- TTLS

EAP- PSK

EAP- IKEv2

PPP 802.3Ethernet 802.11 WLAN Other

Other

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Numerous methods have been defined to work over EAP. The following are commonly supported EAP methods:

■ EAP-TLS (EAP Transport Layer Security): EAP-TLS (RFC 5216) defines how the TLS protocol (described in Chapter 17) can be encapsulated in EAP messages. EAP-TLS uses the handshake protocol in TLS, not its encryption method. Client and server authenticate each other using digital certificates. Client generates a pre-master secret key by encrypting a random number with the server’s public key and sends it to the server. Both client and server use the pre-master to generate the same secret key.

■ EAP-TTLS (EAP Tunneled TLS): EAP-TTLS is like EAP-TLS, except only the server has a certificate to authenticate itself to the client first. As in EAP- TLS, a secure connection (the “tunnel”) is established with secret keys, but that connection is used to continue the authentication process by authenti- cating the client and possibly the server again using any EAP method or legacy method such as PAP (Password Authentication Protocol) and CHAP (Challenge-Handshake Authentication Protocol). EAP-TTLS is defined in RFC 5281.

■ EAP-GPSK (EAP Generalized Pre-Shared Key): EAP-GPSK, defined in RFC 5433, is an EAP method for mutual authentication and session key deri- vation using a Pre-Shared Key (PSK). EAP-GPSK specifies an EAP method based on pre-shared keys and employs secret key-based cryptographic algo- rithms. Hence, this method is efficient in terms of message flows and com- putational costs, but requires the existence of pre-shared keys between each peer and EAP server. The set up of these pairwise secret keys is part of the peer registration, and thus, must satisfy the system preconditions. It provides a protected communication channel when mutual authentication is success- ful for both parties to communicate over and is designed for authentication over insecure networks such as IEEE 802.11. EAP-GPSK does not require any public-key cryptography. The EAP method protocol exchange is done in a minimum of four messages.

■ EAP-IKEv2: It is based on the Internet Key Exchange protocol version 2 (IKEv2), which is described in Chapter 20. It supports mutual authentication and session key establishment using a variety of methods. EAP-TLS is defined in RFC 5106.

EAP Exchanges

Whatever method is used for authentication, the authentication information and authentication protocol information are carried in EAP messages.

RFC 3748 defines the goal of the exchange of EAP messages to be successful authentication. In the context of RFC 3748, successful authentication is an exchange of EAP messages, as a result of which the authenticator decides to allow access by the peer, and the peer decides to use this access. The authenticator’s decision typically involves both authentication and authorization aspects; the peer may successfully authenticate to the authenticator, but access may be denied by the authenticator due to policy reasons.

16.2 / EXTENSIBLE AUTHENTICATION PROTOCOL 525

Figure 16.3 indicates a typical arrangement in which EAP is used. The follow- ing components are involved:

■ EAP peer: Client computer that is attempting to access a network.

■ EAP authenticator: An access point or NAS that requires EAP authentication prior to granting access to a network.

■ Authentication server: A server computer that negotiates the use of a specific EAP method with an EAP peer, validates the EAP peer’s credentials, and authorizes access to the network. Typically, the authentication server is a Remote Authentication Dial-In User Service (RADIUS) server.

The authentication server functions as a backend server that can authenti- cate peers as a service to a number of EAP authenticators. The EAP authentica- tor then makes the decision of whether to grant access. This is referred to as the EAP pass-through mode. Less commonly, the authenticator takes over the role of the EAP server; that is, only two parties are involved in the EAP execution.

As a first step, a lower-level protocol, such as PPP (point-to-point protocol) or IEEE 802.1X, is used to connect to the EAP authenticator. The software entity in the EAP peer that operates at this level is referred to as the supplicant. EAP messages containing the appropriate information for a chosen EAP method are then exchanged between the EAP peer and the authentication server.

EAP messages may include the following fields:

■ Code: Identifies the Type of EAP message. The codes are Request (1), Response (2), Success (3), and Failure (4).

■ Identifier: Used to match Responses with Requests.

■ Length: Indicates the length, in octets, of the EAP message, including the Code, Identifier, Length, and Data fields.

Figure 16.3 EAP Protocol Exchanges

Method

EAP peer/ authenticator

EAP layer

Lower layer

EAP authenticator

EAP layer

Lower layer

Method

EAP peer/ authenticator

EAP layer

Lower layer RADIUS

EAP messages

EAP messages

802.1X, PPP

EAP peer EAP authenticator Authentication server

(RADIUS)

526 CHAPTER 16 / NETWORK ACCESS CONTROL AND CLOUD SECURITY

■ Data: Contains information related to authentication. Typically, the Data field consists of a Type subfield, indicating the type of data carried, and a Type-Data field.

The Success and Failure messages do not include a Data field. The EAP authentication exchange proceeds as follows. After a lower-level

exchange that established the need for an EAP exchange, the authenticator sends a Request to the peer to request an identity, and the peer sends a Response with the identity information. This is followed by a sequence of Requests by the authentica- tor and Responses by the peer for the exchange of authentication information. The information exchanged and the number of Request–Response exchanges needed depend on the authentication method. The conversation continues until either (1) the authenticator determines that it cannot authenticate the peer and transmits an EAP Failure or (2) the authenticator determines that successful authentication has occurred and transmits an EAP Success.

Figure 16.4 gives an example of an EAP exchange. Not shown in the figure is a message or signal sent from the EAP peer to the authenticator using some protocol other than EAP and requesting an EAP exchange to grant network access. One protocol used for this purpose is IEEE 802.1X, discussed in the next section. The first pair of EAP Request and Response messages is of Type identity, in which the authenticator requests the peer’s identity, and the peer returns its claimed identity in the Response message. This Response is passed through the authenticator to the authentication server. Subsequent EAP messages are exchanged between the peer and the authentication server.

Figure 16.4 EAP Message Flow in Pass-Through Mode

EAP peer

EAP-Response/Identity

EAP-Request/Identity

EAP authenticator Authentication server (RADIUS)

EAP-Response/Auth

EAP-Request/Auth

EAP-Response/Auth

EAP-Request/Auth

EAP-Success/Failure

16.3 / IEEE 802.1X PORT-BASED NETWORK ACCESS CONTROL 527

Upon receiving the identity Response message from the peer, the server selects an EAP method and sends the first EAP message with a Type field related to an authentication method. If the peer supports and accepts the selected EAP method, it replies with the corresponding Response message of the same type. Otherwise, the peer sends a NAK, and the EAP server either selects another EAP method or aborts the EAP execution with a failure message. The selected EAP method determines the number of Request/Response pairs. During the exchange the appropriate authentication information, including key material, is exchanged. The exchange ends when the server determines that authentication has succeeded or that no further attempt can be made and authentication has failed.

16.3 IEEE 802.1X PORT-BASED NETWORK ACCESS CONTROL

IEEE 802.1X Port-Based Network Access Control was designed to provide access control functions for LANs. Table 16.1 briefly defines key terms used in the IEEE 802.11 standard. The terms supplicant, network access point, and authentication

Authenticator An entity at one end of a point-to-point LAN segment that facilities authentication of the entity to the other end of the link.

Authentication exchange

The two-party conversation between systems performing an authentication process.

Authentication process

The cryptographic operations and supporting data frames that perform the actual authentication.

Authentication server (AS) An entity that provides an authentication service to an authenticator. This service determines, from the credentials provided by supplicant, whether the supplicant is authorized to access the services provided by the system in which the authenticator resides.

Authentication transport The datagram session that actively transfers the authentication exchange between two systems.

Bridge port A port of an IEEE 802.1D or 802.1Q bridge.

Edge port A bridge port attached to a LAN that has no other bridges attached to it.

Network access port A point of attachment of a system to a LAN. It can be a physical port, such as a single LAN MAC attached to a physical LAN segment, or a logical port, for example, an IEEE 802.11 association between a station and an access point.

Port access entity (PAE) The protocol entity associated with a port. It can support the protocol functionality associated with the authenticator, the supplicant, or both.

Supplicant An entity at one end of a point-to-point LAN segment that seeks to be authenticated by an authenticator attached to the other end of that link.

Table 16.1 Terminology Related to IEEE 802.1X

528 CHAPTER 16 / NETWORK ACCESS CONTROL AND CLOUD SECURITY

server correspond to the EAP terms peer, authenticator, and authentication server, respectively.

Until the AS authenticates a supplicant (using an authentication protocol), the authenticator only passes control and authentication messages between the sup- plicant and the AS; the 802.1X control channel is unblocked, but the 802.11 data channel is blocked. Once a supplicant is authenticated and keys are provided, the authenticator can forward data from the supplicant, subject to predefined access control limitations for the supplicant to the network. Under these circumstances, the data channel is unblocked.

As indicated in Figure 16.5, 802.1X uses the concepts of controlled and uncon- trolled ports. Ports are logical entities defined within the authenticator and refer to physical network connections. Each logical port is mapped to one of these two types of physical ports. An uncontrolled port allows the exchange of protocol data units (PDUs) between the supplicant and the AS, regardless of the authentication state of the supplicant. A controlled port allows the exchange of PDUs between a sup- plicant and other systems on the network only if the current state of the supplicant authorizes such an exchange.

The essential element defined in 802.1X is a protocol known as EAPOL (EAP over LAN). EAPOL operates at the network layers and makes use of an IEEE 802 LAN, such as Ethernet or Wi-Fi, at the link level. EAPOL enables a supplicant to communicate with an authenticator and supports the exchange of EAP packets for authentication.

Figure 16.5 802.1X Access Control

Supplicant

Network access point

Uncontrolled port

Controlled port

Authentication server

Network or Internet

16.4 / CLOUD COMPUTING 529

The most common EAPOL packets are listed in Table 16.2. When the supplicant first connects to the LAN, it does not know the MAC address of the authenticator. Actually it doesn’t know whether there is an authenticator present at all. By sending an EAPOL-Start packet to a special group-multicast address reserved for IEEE 802.1X authenticators, a supplicant can determine whether an authenticator is present and let it know that the supplicant is ready. In many cases, the authenticator will already be notified that a new device has connected from some hardware notification. For example, a hub knows that a cable is plugged in before the device sends any data. In this case the authenticator may preempt the Start mes- sage with its own message. In either case the authenticator sends an EAP-Request Identity message encapsulated in an EAPOL-EAP packet. The EAPOL-EAP is the EAPOL frame type used for transporting EAP packets.

The authenticator uses the EAP-Key packet to send cryptographic keys to the supplicant once it has decided to admit it to the network. The EAP-Logoff packet type indicates that the supplicant wishes to be disconnected from the network.

The EAPOL packet format includes the following fields:

■ Protocol version: version of EAPOL.

■ Packet type: indicates start, EAP, key, logoff, etc.

■ Packet body length: If the packet includes a body, this field indicates the body length.

■ Packet body: The payload for this EAPOL packet. An example is an EAP packet.

Figure 16.6 shows an example of exchange using EAPOL. In Chapter 18, we examine the use of EAP and EAPOL in the context of IEEE 802.11 wireless LAN security.

16.4 CLOUD COMPUTING

There is an increasingly prominent trend in many organizations to move a substan- tial portion of or even all information technology (IT) operations to an Internet- connected infrastructure known as enterprise cloud computing. This section provides an overview of cloud computing. For a more detailed treatment, see [STAL16].

Frame Type Definition

EAPOL-EAP Contains an encapsulated EAP packet.

EAPOL-Start A supplicant can issue this packet instead of waiting for a challenge from the authenticator.

EAPOL-Logoff Used to return the state of the port to unauthorized when the supplicant is finished using the network.

EAPOL-Key Used to exchange cryptographic keying information.

Table 16.2 Common EAPOL Frame Types

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Cloud Computing Elements

NIST defines cloud computing, in NIST SP-800-145 (The NIST Definition of Cloud Computing), as follows:

Figure 16.6 Example Timing Diagram for IEEE 802.1X

EAP peer

EAPOL-Start

EAPOL-EAP (EAP-Request/Identity)

EAPOL-EAP (EAP-Response/Identity)

EAP authenticator Authentication server (RADIUS)

EAPOL-Logoff

EAPOL-EAP (EAP-Response/Auth)

EAPOL-EAP (EAP-Request/Auth)

EAPOL-EAP (EAP-Response/Auth)

EAPOL-EAP (EAP-Request/Auth)

EAPOL-EAP (EAP-Success)

Cloud computing: A model for enabling ubiquitous, convenient, on-demand net- work access to a shared pool of configurable computing resources (e.g., networks, servers, storage, applications, and services) that can be rapidly provisioned and released with minimal management effort or service provider interaction. This cloud model promotes availability and is composed of five essential characteris- tics, three service models, and four deployment models.

The definition refers to various models and characteristics, whose relationship is illustrated in Figure 16.7. The essential characteristics of cloud computing include the following:

■ Broad network access: Capabilities are available over the network and ac- cessed through standard mechanisms that promote use by heterogeneous thin

16.4 / CLOUD COMPUTING 531

or thick client platforms (e.g., mobile phones, laptops, and PDAs) as well as other traditional or cloud-based software services.

■ Rapid elasticity: Cloud computing gives you the ability to expand and reduce resources according to your specific service requirement. For example, you may need a large number of server resources for the duration of a specific task. You can then release these resources upon completion of the task.

■ Measured service: Cloud systems automatically control and optimize resource use by leveraging a metering capability at some level of abstraction appropri- ate to the type of service (e.g., storage, processing, bandwidth, and active user accounts). Resource usage can be monitored, controlled, and reported, provid- ing transparency for both the provider and consumer of the utilized service.

■ On-demand self-service: A consumer can unilaterally provision computing capabilities, such as server time and network storage, as needed automati- cally without requiring human interaction with each service provider. Because the service is on demand, the resources are not permanent parts of your IT infrastructure.

■ Resource pooling: The provider’s computing resources are pooled to serve multiple consumers using a multi-tenant model, with different physical and virtual resources dynamically assigned and reassigned according to consumer demand. There is a degree of location independence in that the customer

Figure 16.7 Cloud Computing Elements

Broad Network Access

Resource Pooling

Rapid Elasticity

E ss

en tia

l C

ha ra

ct er

is tic

s Se

rv ic

e M

od el

s D

ep lo

ym en

t M

od el

s Measured

Service On-Demand Self-Service

Public Private Hybrid Community

Software as a Service (SaaS)

Platform as a Service (PaaS)

Infrastructure as a Service (IaaS)

532 CHAPTER 16 / NETWORK ACCESS CONTROL AND CLOUD SECURITY

generally has no control or knowledge of the exact location of the provided resources, but may be able to specify location at a higher level of abstraction (e.g., country, state, or data center). Examples of resources include storage, processing, memory, network bandwidth, and virtual machines. Even private clouds tend to pool resources between different parts of the same organization.

NIST defines three service models, which can be viewed as nested service alternatives:

■ Software as a service (SaaS): The capability provided to the consumer is to use the provider’s applications running on a cloud infrastructure. The applications are accessible from various client devices through a thin client interface such as a Web browser. Instead of obtaining desktop and server licenses for software products it uses, an enterprise obtains the same functions from the cloud service. SaaS saves the complexity of software installation, maintenance, upgrades, and patches. Examples of services at this level are Gmail, Google’s email service, and Salesforce.com, which helps firms keep track of their customers.

■ Platform as a service (PaaS): The capability provided to the consumer is to deploy onto the cloud infrastructure consumer-created or acquired applica- tions created using programming languages and tools supported by the pro- vider. PaaS often provides middleware-style services such as database and component services for use by applications. In effect, PaaS is an operating system in the cloud.

■ Infrastructure as a service (IaaS): The capability provided to the consumer is to provision processing, storage, networks, and other fundamental computing resources where the consumer is able to deploy and run arbitrary software, which can include operating systems and applications. IaaS enables custom- ers to combine basic computing services, such as number crunching and data storage, to build highly adaptable computer systems.

NIST defines four deployment models:

■ Public cloud: The cloud infrastructure is made available to the general public or a large industry group and is owned by an organization selling cloud ser- vices. The cloud provider is responsible both for the cloud infrastructure and for the control of data and operations within the cloud.

■ Private cloud: The cloud infrastructure is operated solely for an organization. It may be managed by the organization or a third party and may exist on prem- ise or off premise. The cloud provider (CP) is responsible only for the infra- structure and not for the control.

■ Community cloud: The cloud infrastructure is shared by several organizations and supports a specific community that has shared concerns (e.g., mission, security requirements, policy, and compliance considerations). It may be managed by the organizations or a third party and may exist on premise or off premise.

■ Hybrid cloud: The cloud infrastructure is a composition of two or more clouds (private, community, or public) that remain unique entities but are bound together by standardized or proprietary technology that enables data and application portability (e.g., cloud bursting for load balancing between clouds).

Hiva-Network.Com

http://www.hiva-network.com/

16.4 / CLOUD COMPUTING 533

Figure 16.8 illustrates the typical cloud service context. An enterprise main- tains workstations within an enterprise LAN or set of LANs, which are connected by a router through a network or the Internet to the cloud service provider. The cloud service provider maintains a massive collection of servers, which it man- ages with a variety of network management, redundancy, and security tools. In the figure, the cloud infrastructure is shown as a collection of blade servers, which is a common architecture.

Cloud Computing Reference Architecture

NIST SP 500-292 (NIST Cloud Computing Reference Architecture) establishes a reference architecture, described as follows:

Figure 16.8 Cloud Computing Context

Router

Servers

LAN switch

Cloud service

provider Network

or Internet

Router

LAN switch

Enterprise (Cloud user)

The NIST cloud computing reference architecture focuses on the requirements of “what” cloud services provide, not a “how to” design solution and implemen- tation. The reference architecture is intended to facilitate the understanding of the operational intricacies in cloud computing. It does not represent the system architecture of a specific cloud computing system; instead it is a tool for describ- ing, discussing, and developing a system-specific architecture using a common framework of reference.

534 CHAPTER 16 / NETWORK ACCESS CONTROL AND CLOUD SECURITY

NIST developed the reference architecture with the following objectives in mind:

■ to illustrate and understand the various cloud services in the context of an overall cloud computing conceptual model

■ to provide a technical reference for consumers to understand, discuss, catego- rize, and compare cloud services

■ to facilitate the analysis of candidate standards for security, interoperability, and portability and reference implementations

The reference architecture, depicted in Figure 16.9, defines five major actors in terms of the roles and responsibilities:

■ Cloud consumer: A person or organization that maintains a business relation- ship with, and uses service from, cloud providers.

■ Cloud provider: A person, organization, or entity responsible for making a service available to interested parties.

■ Cloud auditor: A party that can conduct independent assessment of cloud services, information system operations, performance, and security of the cloud implementation.

■ Cloud broker: An entity that manages the use, performance, and delivery of cloud services, and negotiates relationships between CPs and cloud consumers.

■ Cloud carrier: An intermediary that provides connectivity and transport of cloud services from CPs to cloud consumers.

The roles of the cloud consumer and provider have already been discussed. To summarize, a cloud provider can provide one or more of the cloud services to meet IT and business requirements of cloud consumers. For each of the three service

Figure 16.9 NIST Cloud Computing Reference Architecture

Cloud consumer

Cloud auditor

Service intermediation

Service aggregation

Service arbitrage

Cloud broker

Cloud provider

Security audit

Performance audit

Privacy impact audit

SaaS Service layer Service orchestration Cloud

service management

PaaS

Hardware

Physical resource layer

Facility

Resource abstraction and control layer

IaaS

Business support

Provisioning/ configuration

Portability/ interoperability

Se cu

ri ty

P ri

va cy

Cloud carrier

16.5 / CLOUD SECURITY RISKS AND COUNTERMEASURES 535

models (SaaS, PaaS, IaaS), the CP provides the storage and processing facilities needed to support that service model, together with a cloud interface for cloud service consumers. For SaaS, the CP deploys, configures, maintains, and updates the operation of the software applications on a cloud infrastructure so that the services are provisioned at the expected service levels to cloud consumers. The consumers of SaaS can be organizations that provide their members with access to software applications, end users who directly use software applications, or software application administrators who configure applications for end users.

For PaaS, the CP manages the computing infrastructure for the platform and runs the cloud software that provides the components of the platform, such as run- time software execution stack, databases, and other middleware components. Cloud consumers of PaaS can employ the tools and execution resources provided by CPs to develop, test, deploy, and manage the applications hosted in a cloud environment.

For IaaS, the CP acquires the physical computing resources underlying the service, including the servers, networks, storage, and hosting infrastructure. The IaaS cloud consumer in turn uses these computing resources, such as a virtual computer, for their fundamental computing needs.

The cloud carrier is a networking facility that provides connectivity and trans- port of cloud services between cloud consumers and CPs. Typically, a CP will set up service level agreements (SLAs) with a cloud carrier to provide services consistent with the level of SLAs offered to cloud consumers, and may require the cloud carrier to provide dedicated and secure connections between cloud consumers and CPs.

A cloud broker is useful when cloud services are too complex for a cloud con- sumer to easily manage. Three areas of support can be offered by a cloud broker:

■ Service intermediation: These are value-added services, such as identity man- agement, performance reporting, and enhanced security.

■ Service aggregation: The broker combines multiple cloud services to meet consumer needs not specifically addressed by a single CP, or to optimize per- formance or minimize cost.

■ Service arbitrage: This is similar to service aggregation except that the services being aggregated are not fixed. Service arbitrage means a broker has the flexibil- ity to choose services from multiple agencies. The cloud broker, for example, can use a credit-scoring service to measure and select an agency with the best score.

A cloud auditor can evaluate the services provided by a CP in terms of secu- rity controls, privacy impact, performance, and so on. The auditor is an independent entity that can assure that the CP conforms to a set of standards.

16.5 CLOUD SECURITY RISKS AND COUNTERMEASURES

In general terms, security controls in cloud computing are similar to the security controls in any IT environment. However, because of the operational models and technologies used to enable cloud service, cloud computing may present risks that are specific to the cloud environment. The essential concept in this regard is that the enterprise loses a substantial amount of control over resources, services, and applications but must maintain accountability for security and privacy policies.

536 CHAPTER 16 / NETWORK ACCESS CONTROL AND CLOUD SECURITY

The Cloud Security Alliance [CSA10] lists the following as the top cloud- specific security threats, together with suggested countermeasures:

■ Abuse and nefarious use of cloud computing: For many CPs, it is relatively easy to register and begin using cloud services, some even offering free limited trial periods. This enables attackers to get inside the cloud to conduct various attacks, such as spamming, malicious code attacks, and denial of service. PaaS providers have traditionally suffered most from this kind of attacks; however, recent evidence shows that hackers have begun to target IaaS vendors as well. The burden is on the CP to protect against such attacks, but cloud service cli- ents must monitor activity with respect to their data and resources to detect any malicious behavior.

Countermeasures include (1) stricter initial registration and valida- tion processes; (2) enhanced credit card fraud monitoring and coordination; (3) comprehensive introspection of customer network traffic; and (4) monitor- ing public blacklists for one’s own network blocks.

■ Insecure interfaces and APIs: CPs expose a set of software interfaces or APIs that customers use to manage and interact with cloud services. The security and availability of general cloud services are dependent upon the security of these basic APIs. From authentication and access control to encryption and activity monitoring, these interfaces must be designed to protect against both accidental and malicious attempts to circumvent policy.

Countermeasures include (1) analyzing the security model of CP inter- faces; (2) ensuring that strong authentication and access controls are imple- mented in concert with encrypted transmission; and (3) understanding the dependency chain associated with the API.

■ Malicious insiders: Under the cloud computing paradigm, an organization relinquishes direct control over many aspects of security and, in doing so, con- fers an unprecedented level of trust onto the CP. One grave concern is the risk of malicious insider activity. Cloud architectures necessitate certain roles that are extremely high risk. Examples include CP system administrators and managed security service providers.

Countermeasures include the following: (1) enforce strict supply chain management and conduct a comprehensive supplier assessment; (2) specify human resource requirements as part of legal contract; (3) require transpar- ency into overall information security and management practices, as well as compliance reporting; and (4) determine security breach notification processes.

■ Shared technology issues: IaaS vendors deliver their services in a scalable way by sharing infrastructure. Often, the underlying components that make up this infrastructure (CPU caches, GPUs, etc.) were not designed to offer strong iso- lation properties for a multi-tenant architecture. CPs typically approach this risk by the use of isolated virtual machines for individual clients. This approach is still vulnerable to attack, by both insiders and outsiders, and so can only be a part of an overall security strategy.

Countermeasures include the following: (1) implement security best practices for installation/configuration; (2) monitor environment for unauthor- ized changes/activity; (3) promote strong authentication and access control

16.6 / DATA PROTECTION IN THE CLOUD 537

for administrative access and operations; (4) enforce SLAs for patching and vulnerability remediation; and (5) conduct vulnerability scanning and configuration audits.

■ Data loss or leakage: For many clients, the most devastating impact from a security breach is the loss or leakage of data. We address this issue in the next subsection.

Countermeasures include the following: (1) implement strong API ac- cess control; (2) encrypt and protect integrity of data in transit; (3) analyze data protection at both design and run time; and (4) implement strong key generation, storage and management, and destruction practices.

■ Account or service hijacking: Account or service hijacking, usually with stolen credentials, remains a top threat. With stolen credentials, attackers can often access critical areas of deployed cloud computing services, allowing them to compromise the confidentiality, integrity, and availability of those services.

Countermeasures include the following: (1) prohibit the sharing of account credentials between users and services; (2) leverage strong two- factor authentication techniques where possible; (3) employ proactive monitor- ing to detect unauthorized activity; and (4) understand CP security policies and SLAs.

■ Unknown risk profile: In using cloud infrastructures, the client necessarily cedes control to the CP on a number of issues that may affect security. Thus the client must pay attention to and clearly define the roles and responsibili- ties involved for managing risks. For example, employees may deploy applica- tions and data resources at the CP without observing the normal policies and procedures for privacy, security, and oversight.

Countermeasures include (1) disclosure of applicable logs and data; (2) partial/full disclosure of infrastructure details (e.g., patch levels and firewalls); and (3) monitoring and alerting on necessary information.

Similar lists have been developed by the European Network and Information Security Agency [ENIS09] and NIST [JANS11].

16.6 DATA PROTECTION IN THE CLOUD

As can be seen from the previous section, there are numerous aspects to cloud security and numerous approaches to providing cloud security measures. A further example is seen in the NIST guidelines for cloud security, specified in SP-800-14 and listed in Table 16.3. Thus, the topic of cloud security is well beyond the scope of this chapter. In this section, we focus on one specific element of cloud security.

There are many ways to compromise data. Deletion or alteration of records without a backup of the original content is an obvious example. Unlinking a record from a larger context may render it unrecoverable, as can storage on unreliable media. Loss of an encoding key may result in effective destruction. Finally, unau- thorized parties must be prevented from gaining access to sensitive data.

538 CHAPTER 16 / NETWORK ACCESS CONTROL AND CLOUD SECURITY

Governance Extend organizational practices pertaining to the policies, procedures, and standards used for application development and service provisioning in the cloud, as well as the design, implementation, testing, use, and monitoring of deployed or engaged services.

Put in place audit mechanisms and tools to ensure organizational practices are followed throughout the system life cycle.

Compliance Understand the various types of laws and regulations that impose security and privacy obligations on the organization and potentially impact cloud computing initiatives, particularly those involving data location, privacy and security controls, records management, and electronic discovery requirements.

Review and assess the cloud provider’s offerings with respect to the organizational requirements to be met and ensure that the contract terms adequately meet the requirements.

Ensure that the cloud provider’s electronic discovery capabilities and processes do not compromise the privacy or security of data and applications.

Trust Ensure that service arrangements have sufficient means to allow visibility into the security and privacy controls and processes employed by the cloud provider, and their performance over time.

Establish clear, exclusive ownership rights over data. Institute a risk management program that is flexible enough to adapt to the constantly evolving and

shifting risk landscape for the life cycle of the system. Continuously monitor the security state of the information system to support ongoing risk management

decisions.

Architecture Understand the underlying technologies that the cloud provider uses to provision services, including the implications that the technical controls involved have on the security and privacy of the system, over the full system life cycle and across all system components.

Identity and access management Ensure that adequate safeguards are in place to secure authentication, authorization, and other identity and access management functions, and are suitable for the organization.

Software isolation Understand virtualization and other logical isolation techniques that the cloud provider employs in its multi-tenant software architecture, and assess the risks involved for the organization.

Data protection Evaluate the suitability of the cloud provider’s data management solutions for the organizational data concerned and the ability to control access to data, to secure data while at rest, in transit, and in use, and to sanitize data.

Take into consideration the risk of collating organizational data with those of other organizations whose threat profiles are high or whose data collectively represent significant concentrated value.

Fully understand and weigh the risks involved in cryptographic key management with the facilities available in the cloud environment and the processes established by the cloud provider.

Availability Understand the contract provisions and procedures for availability, data backup and recovery, and disaster recovery, and ensure that they meet the organization’s continuity and contingency planning requirements.

Ensure that during an intermediate or prolonged disruption or a serious disaster, critical operations can be immediately resumed, and that all operations can be eventually reinstituted in a timely and organized manner.

Incident response Understand the contract provisions and procedures for incident response and ensure that they meet the requirements of the organization.

Table 16.3 NIST Guidelines on Security and Privacy Issues and Recommendations

16.6 / DATA PROTECTION IN THE CLOUD 539

Ensure that the cloud provider has a transparent response process in place and sufficient mechanisms to share information during and after an incident.

Ensure that the organization can respond to incidents in a coordinated fashion with the cloud provider in accordance with their respective roles and responsibilities for the computing environment.

Table 16.3 Continued

The threat of data compromise increases in the cloud, due to the number of and interactions between risks and challenges that are either unique to the cloud or more dangerous because of the architectural or operational characteristics of the cloud environment.

Database environments used in cloud computing can vary significantly. Some providers support a multi-instance model, which provides a unique DBMS running on a virtual machine instance for each cloud subscriber. This gives the subscriber complete control over role definition, user authorization, and other administrative tasks related to security. Other providers support a multi-tenant model, which pro- vides a predefined environment for the cloud subscriber that is shared with other tenants, typically through tagging data with a subscriber identifier. Tagging gives the appearance of exclusive use of the instance, but relies on the CP to establish and maintain a sound secure database environment.

Data must be secured while at rest, in transit, and in use, and access to the data must be controlled. The client can employ encryption to protect data in transit, though this involves key management responsibilities for the CP. The client can enforce access control techniques but, again, the CP is involved to some extent depending on the service model used.

For data at rest, the ideal security measure is for the client to encrypt the data- base and only store encrypted data in the cloud, with the CP having no access to the encryption key. So long as the key remains secure, the CP has no ability to read the data, although corruption and other denial-of-service attacks remain a risk.

A straightforward solution to the security problem in this context is to encrypt the entire database and not provide the encryption/decryption keys to the service provider. This solution by itself is inflexible. The user has little ability to access individual data items based on searches or indexing on key parameters, but rather would have to download entire tables from the database, decrypt the tables, and work with the results. To provide more flexibility, it must be possible to work with the database in its encrypted form.

An example of such an approach, depicted in Figure 16.10, is reported in [DAMI05] and [DAMI03]. A similar approach is described in [HACI02]. Four enti- ties are involved:

■ Data owner: An organization that produces data to be made available for controlled release, either within the organization or to external users.

■ User: Human entity that presents requests (queries) to the system. The user could be an employee of the organization who is granted access to the data- base via the server, or a user external to the organization who, after authenti- cation, is granted access.

■ Client: Frontend that transforms user queries into queries on the encrypted data stored on the server.

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■ Server: An organization that receives the encrypted data from a data owner and makes them available for distribution to clients. The server could in fact be owned by the data owner but, more typically, is a facilit

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