Question

Bob and Alice are using the Diffie-Hellman key exchange method to share a secret private key K (for example, to be used in a private key cryptosystem such as a shift cipher). They have decided to use p=23 and α=5. Both p and α are known to the public. Suppose Alice chooses x = 3 and Bob chooses y = 6 as their private keys. Find what each of them sends to the other person.

Answer 1

Sure, let's calculate what Alice and Bob will each send to each other.

First, let's revisit the formula we are using:
We use Alice's private key (x) and Bob's private key (y), and the publicly known alpha value (α) and prime number p.

Alice computes `(α^x) mod p` and sends this to Bob,
Bob computes `(α^y) mod p` and sends this to Alice.

Given the values are:
Alpha (α) = 5
Prime number (p) = 23
Alice's private key (x) = 3
Bob's private key (y) = 6

We substitute these values into the formulas:

1. Compute what Alice sends to Bob: `(α^x) mod p` which is `(5^3) mod 23`. After computing this, we find that Alice sends the value `10` to Bob.

2. Compute what Bob sends to Alice: `(α^y) mod p` which is `(5^6) mod 23`. After computation, we find that Bob sends the value `8` to Alice.

So, in conclusion, Alice sends `10` to Bob and Bob sends `8` to Alice for the establishment of the shared private key.