Suppose Alice, Bob, and Carol want to use secret keys to authenticate each other. If they all used the same secret key K, then Bob could impersonate Carol to Alice (actually any of the three can impersonate the other to the third). Suppose instead that each had their own secret key, so Alice uses KA, Bob uses KB, and Carol uses KC. This means that each one, to prove his/her identity (authenticate), responds to a challenge with a function of his/her secret key and the challenge. Is this more secure that having them all use the same secret key K? (Hint: what does Alice need to know in order to verify Carols answer to Alices challenge?) Please write a theory answer.