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Understanding Hill Ciphers and Modular Matrix Inverses in Cryptography

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In today's session, we focused on the Hill cipher and its historical significance in cryptography. We discussed critical concepts like diffusion and confusion in block ciphers, emphasizing how changes in plaintext affect the ciphertext. The Hill cipher employs linear algebra, using an invertible matrix for encryption and decryption modulo 26. We explored matrix operations and demonstrated examples to illustrate the process. Additionally, we addressed techniques for breaking ciphers via known plaintext. For further questions, feel free to reach out.

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Understanding Hill Ciphers and Modular Matrix Inverses in Cryptography

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  1. DTTF/NB479: Dszquphsbqiz Day 7 • Announcements: • We’ll attempt to grade Friday’s quizzes by end of day on Friday and leave outside my office • Matlabtutorial linked to in syllabus • Questions? • Today: • Modular matrix inverses • Hill ciphers

  2. Q1 Block Ciphers • So far, changing 1 character in the plaintext changes ___ character(s) in the ciphertext. • Shannon outlined qualities of good ciphers: • Diffusion: Changing one character of the plaintext changes _____ characters in the ciphertext • Makes frequency analysis much tougher! • Confusion: Each character of the ciphertext interacts with several parts of the key • Block ciphershave both qualities: • DES (64 bits), AES (128 bits), Hill ciphers (smaller; today)

  3. Q2 Hill Ciphers • Lester Hill, 1929. Not used much, but is historically significant: first time linear algebra used in crypto • Use an n x n matrix M. Encrypt by breaking plaintext into blocks of length n (padding with x’s if needed) and multiplying each by M (mod 26). • Example: Encrypt “hereissomeonetoencrypt” using M • her eissom eon eto enc ryptxx • (7, 4, 17) (4, 8, 18) … (19, 23, 23) • (2, 5, 25) (0, 2, 22) … (0, 22, 15) • cfzacwygavnsaveancsddawp • “CFZACWYGAVNSAVEANCSDDAWP”

  4. Q3 Decrypting • Reverse the process, multiplying each block by M inverse (mod n) • Theorem: If a matrix M is invertible mod n, then gcd(det(M), n) = 1 • Proof on board

  5. Q4-Q6 Modular matrix inverse (§3.8) • The Hill cipher requires us to invert a matrix mod 26. • For a 2x2 matrix, this is easy. • Many numerical packages allow us to invert a matrix, but using floating point numbers. • How do we combine the two? • Demo of my code

  6. Q7-Q8 How to break via known plaintext? • Answering Q7 preps you to do 2.13 #14 on HW2 if you want to earn an early day • You may leave when done

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