### Instruction: Two users, Alice and Bob, want to communicate securely over an insecure network. They agree on a prime number p=23 and a primitive root of p g=5. Alice chooses a secret key a=6 and shares g^a mod p= 8 with Bob. Bob chooses a secret key b= 15 and shares g^b mod p=19 with Alice. What is the shared secret key that Alice and Bob will use to encrypt their messages to each other?### Response: To find the shared secret key, both Alice and Bob will use each other's public keys and their own private keys. 

For Alice, she will compute (g^b mod p)^a mod p, which is (19^6) mod 23.
For Bob, he will compute (g^a mod p)^b mod p, which is (8^15) mod 23.

Both of these calculations should result in the same shared secret key.

(19^6) mod 23 = 2
(8^15) mod 23 = 2

So, the shared secret key that Alice and Bob will use to encrypt their messages to each other is 2.