We quickly talked about the Lorenz cipher (chapter 6 contains a bit of material, much more is available on Tony Sale's page on the Lorenz cipher), and continued with modern cryptography, and the basics of public-key cryptography. In particular, we saw Diffie-Hellman key-exchagen (see the book). We also covered some modular arithmetic (including Fermat's Little Theorem).

If you need more material on modular arithmetic, check

• a very gentle introduction to clock arithmetic
• the Wikipedia article on modular arithmetic
• modular arithmetic from a more mathematical point of view
• more on modular arithmetic and number theory

1. (Menus) You have located the crib "ninteenhundredandfortyfive" in a piece of ciphertext.

You want to find loops and construct a menu. For example here is a loop of length 3:

      25
   I  ->  R
   \     /
 2  \   /  21
     v v
      T
   

In word: I turns to T (2), R turns to T (21) I turns to R (25).

Construct a menu including this and at least three other loops (there are multiple loops of lengths 3 and 4). Write the menu in the style of class, with the positions next to the arrows, and the arrows pointing from the plaintext letter to the ciphertext letter.

2. (Modular Arithmetic) Compute the following numbers:

• 6 + 11 + 14 + 3 (mod 7)

• 7 * 5 * 13 (mod 12)

• 7 + (10 * 9)  (mod 11)

3. (Diffie-Hellman) Suppose Alice and Bob have agreed on the modolus 13 for the Diffie-Hellman key exchange.

• Find a good base (that is, a base that generates all numbers from 1-12 when going through its powers).

• Find a bad base (other than 0 and 1) (that is, a base that does not generate all of 1-12).

4. (Diffie-Hellman) Suppose Alice and Bob have agreed on the modulus 17, and the base d = 3. Alice uses as her secret function exponentiation with a = 5, Bob uses as his secret function exponentiation to the b = 7th power. Show how Alice and Bob arrive at their common secret, that is, what they compute, what the communicate, and what they end up with.