Mathematical Concepts in Cryptography 355

1. Mathematical Concepts in Cryptography 355 Brian Tafel, Mark Shippy Raymond Fuller, and Mireya

2. The Hill Cipher uses linear algebra to disguise frequency patterns that occur in natural language. A. In the forward direction or encryption process we use matrix multiplication. B. In the reverse or decryption process we use matrix inversion. C. First, let’s learn a little about linear algebra.

3. Matrix Multiplication 2x + 3y – 4z = 5 -2x + z = 7 3x + 2y + 2z = 3 Linear system | x | X = | y | | z | | 5 | B = | 7 | | 3 | | 2 3 -4 | | 5 | A = | -2 0 1 | | 7 | | 3 2 2 | | 3 | Linear system in matrix form AX = B

4. Matrix Multiplication Key matrix GYBNQKURP = 6,24,1,13,16,10,20,17,15 | 57 | = | 122 | | 169 | | 15| = |14| MOD 26 | 7 | | 6 24 1 | | 0 | | 13 16 10 | | 2 | | 20 17 15| | 19 | ACT A = 0, C = 2, T = 19 Becomes the cipher text POH P = 15, O = 14, H = 7 ( 6)(0) + (24)(2) + ( 1)(19) = 57 (13)(0) + (16)(2) + (10)(19) = 122 (20)(0) + (17)(2) + (15)(19) = 169

5. Matrix Decryption First you take the inverse of the Key matrix, which is too horrible to explain. This gives you IFKVIVVMI = 8,5,10,21,8,21,21,12,8 Using are Encrypted cipher text POH = 15,14,7 | 8 5 10 | | 15 | | 21 8 21 | | 14 | |21 12 8 | | 7 | | 260 | = | 574 | | 539| | 0 | = | 2| mod 26 |19| Which gives us are message 0,2,19 or ACT

6. Use of Determinants If the determinate is zero, or has factors like 2 and 13 then we cannot use Hill Cipher. One variant uses 3 extra symbols to make the modulus 29 (prime). = a b – c d | a b | | c d | 2 X 2 Matrix | a b c | | d e f | | g h j | = a(ej – hf) – b(dj – gf) + c(dh – ge) 3 X 3 Matrix

7. Mechanical Implementation Hill and a partner were awarded a patent for a 6 x 6 matrix mod 26 machine using gears and chains. Theories such as Hill’s became automated during the Industrial Revolution. In 1927 this led to the Enigma machine.

8. Drawbacks of Hill cipher The Linearity of the cipher is its downfall. The know Plaintext attack is where the attacker Has samples of both the Plaintext and Encrypted versions. During WWII the Germans would give a weather forecast every day at the same time. The message would contain the word wetter in the same location in every message. Knowing the local weather also helped guess the other parts of the message.

9. Question ???? The Hill Cipher utilizes What type of complex mathematics?

10. Answer Linear Algebra