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An Introduction to Hill Ciphers

An Introduction to Hill Ciphers. Using Linear Algebra. Brian Worthington University of North Texas MATH 2700.002 5/10/2010. Hill Ciphers. Created by Lester S. Hill in 1929 Polygraphic Substitution Cipher Uses Linear Algebra to Encrypt and Decrypt. Simple Substitution Ciphers.

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An Introduction to Hill Ciphers

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  1. An Introduction to Hill Ciphers Using Linear Algebra Brian Worthington University of North Texas MATH 2700.002 5/10/2010

  2. Hill Ciphers • Created by Lester S. Hill in 1929 • Polygraphic Substitution Cipher • Uses Linear Algebra to Encrypt and Decrypt

  3. Simple Substitution Ciphers • Work by substituting one letter with another letter. • Easy to crack using Frequency Analysis

  4. Letter to Letter Substitution Unencrypted = HELLO WORLD Encrypted = ITSSG VKGSR

  5. Polygraphic Substitution Ciphers • Encrypts letters in groups • Frequency analysis more difficult

  6. Hill Ciphers • Polygraphic substitution cipher • Uses matrices to encrypt and decrypt • Uses modular arithmetic (Mod 26)

  7. Modular Arithmetic • For a Mod b, divide a by b and take the remainder. • 14 ÷ 10 = 1 R 4 • 14 Mod 10 = 4 • 24 Mod 10 = 4

  8. Modulus Theorem

  9. Modulus Examples

  10. Modular Inverses • Inverse of 2 is ½ (2 · ½ = 1) • Matrix Inverse: AA-1= I • Modular Inverse for Mod m: (a · a-1) Mod m = 1 • For Modular Inverses, a and m must NOT have any prime factors in common

  11. Modular Inverses of Mod 26 Example – Find the Modular Inverse of 9 for Mod 26 9 · 3 = 27 27 Mod 26 = 1 3 is the Modular Inverse of 9Mod26

  12. Hill Cipher Matrices • One matrix to encrypt, one to decrypt • Must be n x n, invertible matrices • Decryption matrix must be modular inverse of encryption matrix in Mod 26

  13. Modularly Inverse Matrices • Calculate determinant of first matrix A, det A • Make sure that det A has a modular inverse for Mod 26 • Calculate the adjugate of A, adj A • Multiply adj A by modular inverse of det A • Calculate Mod 26 of the result to get B • Use A to encrypt, B to decrypt

  14. Modular Reciprocal Example

  15. Encryption • Assign each letter in alphabet a number between 0 and 25 • Change message into 2 x 1 letter vectors • Change each vector into 2 x 1 numeric vectors • Multiply each numeric vector by encryption matrix • Convert product vectors to letters

  16. Letter to Number Substitution

  17. Change Message to Vectors Message to encrypt = HELLO WORLD

  18. Multiply Matrix by Vectors

  19. Convert to Mod 26

  20. Convert Numbers to Letters HELLO WORLD has been encrypted to SLHZY ATGZT

  21. Decryption • Change message into 2 x 1 letter vectors • Change each vector into 2 x 1 numeric vectors • Multiply each numeric vector by decryption matrix • Convert new vectors to letters

  22. Change Message to Vectors Message to encrypt = SLHZYATGZT

  23. Multiply Matrix by Vectors

  24. Convert to Mod 26

  25. Convert Numbers to Letters SLHZYATGZT has been decrypted to HELLO WORLD

  26. Conclusion • Creating valid encryption/decryption matrices is the most difficult part of Hill Ciphers. • Otherwise, Hill Ciphers use simple linear algebra and modular arithmetic

  27. Questions?

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