Perfect zero-knowledge languages can be recognized in two rounds

Abstract

A hierarchy of probabilistic complexity classes generalizing NP has recently emerged in the work of [Ba], [GMR], and [GS]. The IP hierarchy is defined through the notion of an interactive proof system, in which an all powerful prover tries to convince a probabilistic polynomial time verifier that a string w is in a language L. The verifier tosses coins and exchanges messages back and forth with the prover before he decides whether to accept w. This proof-system yields "probabilistic" proofs: the verifier may erroneously accept or reject w with small probability. In [GMR] such a protocol was defined to be a zero-knowledge protocol if at the end of the interaction the verifier has learned nothing except that w ∈ L. We study complexity theoretic implications of a language having this property. In particular we prove that if L admits a zeroknowledge proof then L can also be recognized by a two round interactive proof. This complements a result by Fortnow [F] where it is proved that the complement of L has a two round interactive proof protocol. The methods of proof are quite similar to those of Fortnow [F]. As in his case the proof works under the assumption that the original protocol is only zero-knowledge with respect to a specific verifier.