Network security: authentication

1) public key algorithm: suppose Alice's public key is A1, private key is A2, Bob's public key is B1, private key is B2. For Alice to authentication Bob, she can challenge Bob by sending a nonce, what does Bob send back? When Bob sends back to Alice a session key, what key bob uses for encrypting the session key.

2) Secret key algorithm: suppose Alice, Bob shares keys A, B with an authentication server S, respectively. For S to authentication Alice, Alice has to send a nonce to S encrypted with which key? S also provide Alice with a message to be sent to Bob containing Alices identity and a session key, which key is used to encrypt the message?