secret key technology

Suppose Alice, Bob, and Carol want to use secret key technology 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 could 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 or her identity, responds to a challenge with a function of his or her secret key. Is this more secure than having them all use the same secret key? What does Alice need to know in order to verify Carol’s answer to Alice’s challenge?

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