390 Codes, Ciphers, and Cryptography

1. 390 Codes, Ciphers, and Cryptography Polygraphic Substitution Ciphers – Hill’s System

2. Hill’s System • We now look at the system for enciphering blocks of text developed by Lester Hill. • Matrices form the basis of this substitution cipher! • We’ll work with blocks of size two letters – the idea can be generalized to larger blocks.

3. Steps to Encipher a Message • 1. Choose a 2 x 2 matrix with entries in Z26 for a key. • Make sure that (ad – bc)-1 (mod 26) exists, i.e. (ad – bc) = 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, or 25. • This will guarantee that A-1 exists (mod 26).

4. Steps to Encipher a Message • 2. Split the plaintext into pairs and assign numbers to each plaintext letter, with a = 1, b = 2, … , z = 26 = 0 (mod 26). • Plaintext: p1p2|p3p4| … |pn-1pn • If necessary, append an extra character to the plaintext to get an even number of plaintext characters.

5. Steps to Encipher a Message

6. Example 8 • Use Hill’s scheme to encipher the message: “Meet me at the usual place at ten rather than eight o’clock.”

7. Solution: For the key, choose a 2 x 2 matrix, with entries in Z26. Note that (ad – bc) (mod 26) = (9*7 – 4*5) (mod 26) = (63 – 20) (mod 26) = 43 (mod 26) = 17 (mod 26) and 17-1 (mod 26) exists! More on this later … Example 8

8. Example 8 • Next convert the plaintext into pairs of numbers from Z26: • me | et | me | at … cl | oc | kz. • 13,5 | 5,20 | 13,5 | 1,20 | … 3,12 | 15,3 | 11,0 • Now convert the plaintext to numbers to ciphertext numbers, using (*) above.

9. Example 8

10. Example 8 • Thus, “me” is encrypted as “GV”. • Try the next pair!

11. Example 8

12. Example 8 • Thus, “et” is encrypted as “UI”. • HW – Finish encrypting message! • Note that for the word “meet”, the first “e” is encrypted as “G” and the second “e” is encrypted as “U”. • Frequency analysis won’t work for this scheme!

13. Deciphering a Message • To decipher a message encrypted with Hill’s Scheme, we can use the idea of matrix inverses! • Since ciphertext (ck,ck+1) is obtained from plaintext (pk,pk+1) by multiplying key matrix A by plaintext (pk,pk+1), all we need to do is multiply matrix A-1 by ciphertext (ck,ck+1).

14. Deciphering a Message

15. Deciphering a Message

16. Deciphering a Message • The same idea will work for matrices of numbers from Z26! • Matrix A will be invertible, provided that (ad-bc)-1 (mod 26) exists! • The only difference is that instead of 1/(ad-bc), we need to use (ad-bc)-1.

17. Deciphering a Message

18. Deciphering a Message

19. Deciphering a Message

20. Example 9 • Decipher the ciphertext found above in Example 8! • Write ciphertext as pairs of numbers in Z26: • GV | UI • 7,22 | 21,9 • Use the inverse of the key matrix to decipher!

21. Example 9

22. Example 9 • Thus, “GV” is deciphered as “me”. • Repeat with “UI”.

23. Example 9

24. Example 9 • Thus, “UI” is deciphered as “et”.

25. References • Cryptological Mathematics by Robert Edward Lewand (section on matrices).