Statistical Zero-Knowledge Languages can be Recognized in Two Rounds

Recently, a hierarchy of probabilistic complexity classes generalizing NP has emerged in the work of Babai [B], and Goldwasser, Micali, and Racko [GMR1], and Goldwasser and Sipser [GS]. The class IP is de ned through the computational model of an interactive prover-veri er pair. Both Turing machines in a pair receive a common input and exchange messages. Every move of the veri er as well as its nal determination of whether to accept or reject w are the result of random polynomial time computations on the input and all messages sent so far. The prover has no resource bounds. A language, L, is in IP if there is a prover-veri er pair such that: 1.) when w 2 L, the veri er accepts with probability at least 1 2 jwj and, 2.) when w 62 L, the veri er interacting with any prover accepts with probability at most 2 jwj. Such a prover-veri er pair is called an interactive proof for L. In addition to de ning interactive proofs, Goldwasser, Micali, and Racko [GMR1] further de ned zero-knowledge interactive proofs. Informally, an interacting pair is a zero-knowledge proof for a language, L, if it is an interactive proof for L with the additional constraint that the prover reveals \nothing" (except language membership) to any veri er that the veri er could not have computed itself. There are three formal de nitions for \nothing" which lead to three types of zero-knowledge: perfect, statistical, and computational, each more restrictive than the next. We show that the rst and second de nitions are very restrictive. Speci cally, any language, L, that has a statistical zero-knowledge interactive proof with an unbounded number of interactions has an interactive proof with two interactions. This complements a result by Fortnow [F] who showed that under the same hypothesis, the complement of L has an interactive proof with two interactions. Warning: Essentially this paper has been published in Journal of Computer and System Sciences and is hence subject to copyright restrictions. It is for personal use only.