This problem illustrates the point that the Diffie-Hellman protocol is not secure without the step where you take the modulus; i.e. the "Indiscrete Log Problem" is not a hard problem! You are Eve and have captured Alice and Bob and imprisoned them. You overhear the following dialog.

Bob: Oh, let's not bother with the prime in the Diffie-Hellman protocol, it will make things easier.

Alice: Okay, but we still need a base \(\alpha\) to raise things to. How about \(g=3\) ?

Bob: All right, then my result is 27.

Alice: And mine is 243.

What is Bob's secret \(X_{B}\) and Alice's secret \(X_{A}\) ? What is their secret combined key? (Don't forget to show your work.)