Problem 1.

Solve the following modular arithmetic problems:

What is 36 modulo 6?

What is 32 modulo 7?

What are 318 and 968 modulo 9?

What is 318 968 modulo 9? (Hint: dont multiply 318 and 968 directly)

What is 96711 modulo 9? (Hint: again, dont compute 96711 directly

Problem 2.

Alice and Bob want to communicate with each other, and they agree to use Diffie-Hellman with prime p = 13 and generator g = 2.

(a) Alice picks a = 3 as her private key. What is her public key?

(b) Bob picks b = 7 as his private key. What is his public key?

(c) What is their shared secret key s? Show how Alice would compute it and how Bob would compute it.

Problem 3

To generate a shared secret key using a public channel, Alice and Bob decided to use Diffie-Hellman algorithm they also agreed on p = 23 and g = 5.

Alice chooses a = 6 and sends 5⁶ mod 23 = 8.

Bob chooses b = 15 and sends 19.

What is the shared secret? Show work.

Problem 4:

Alice and Bob decide to communicate. They want to establish a shared secret key using the Diffie-Hellmanprotocol. First, they agree to use P = 13 and G = 7. Alice then chooses A as her secret key and sends Bob (Al = G^Amod p = 8). Bob chooses 3 as his secret integer and sends Alice (Bo = G³ = 5 mod p).

What is a shared secret between Alice and Bob. Show work.