Cryptography General Question:

Informally, an interactive proof is a protocol between two parties in which one party, called the prover, tries to prove a certain fact to the other party, called the verifier. An interactive proof usually takes the form of a challenge-response protocol, in which the prover and the verifier exchange messages and the verifier outputs either "accept" or "reject" at the end of the protocol. Besides their theoretical interests, interactive proofs have found applications in cryptography and computer security such as identification and authentication. In these situations, the fact to be proved is usually related to the prover's identity, e.g., the prover's private key.

The following properties of interactive proofs are quite useful, especially in cryptographic applications:

Completeness: The verifier always accepts the proof if the prover knows the fact and both the prover and the verifier follow the protocol.
Soundness: The verifier always rejects the proof if the prover does not know the fact, as long as the verifier follows the protocol.
Zero knowledge: The verifier learns nothing about the fact being proved (except that it is correct) from the prover that he could not already learn without the prover. In a zero-knowledge proof, the verifier cannot even later prove the fact to anyone else.

A typical round in a zero-knowledge proof consists of a "commitment" message from the prover, followed by a challenge from the verifier, and then a response to the challenge from the prover. The protocol may be repeated for many rounds. Based on the prover's responses in all the rounds, the verifier decides whether to accept or reject the proof.