Password security: explore password cracking and suggesting a strong password

Not so much a research paper, but a mathematical exploration. It is an International Baccalaureate mathematics higher level internal assessment and has very specific guidelines, which are on pages 64-70 of the attached guide. The exploration should be mathematically advanced (preferably at the level of the IB mathematics HL syllabus p10-56 in the attached guide). The actual writing should be direct. Keep the BS to the minimum. Should include an introduction and a conclusion+reflection. Should be 12 pages in length, excluding the bibliography. Do not make a title page, just have a title at the top of the first page. Only the citations should be MLA, the rest of the formatting should be utilized to condense as much information and exploration as possible within the 12 pages that are allowed. Diagrams would be great where seen helpful or complementary. The topic itself can be modified if the writer believes that the work can be done better with a modified topic. The most important part is to follow and make the best work according to the guidelines in the attachment