Diffie-Hellman Key Exchange Example: Finding Private and Shared Keys

To find A's private key Xa, we need to solve the equation:

9 = 2^Xa mod 11

We can manually calculate the powers of 2 modulo 11:

2^1 mod 11 = 2 2^2 mod 11 = 4 2^3 mod 11 = 8 2^4 mod 11 = 5 2^5 mod 11 = 10 2^6 mod 11 = 9 2^7 mod 11 = 7 2^8 mod 11 = 3 2^9 mod 11 = 6 2^10 mod 11 = 1

Therefore, we can see that 2^6 mod 11 = 9, which means A's private key Xa is 6.

To find the shared secret key K, we need to use the following formula:

K = Yb^Xa mod q

Substituting the given values, we get:

K = 3^6 mod 11

Again, we can manually calculate the powers of 3 modulo 11:

3^1 mod 11 = 3 3^2 mod 11 = 9 3^3 mod 11 = 5 3^4 mod 11 = 4 3^5 mod 11 = 1

Therefore, 3^6 mod 11 = 9, which means the shared secret key K is 9.