Alice and bob are on two sides of a river. the only way to send a message from one side to the other is to send it via eve, who operates a ferry. alice and bob use public key crypto and have their public keys displayed in billboards beside the riverbank which can be seen from the other side. suppose that alice wants to start communicating securely with bob. however, alice and bob are worried that malicious eve can eavesdrop on their communications and change it.

design an efficient scheme that would allow alice and bob to communicate confidentially communicate, such that

(a) eve cannot read the messages
(b) eve cannot replay messages, and
(c) eve cannot modify messages.