Composition of Zero-Knowledge Proofs with Efficient Provers

Abstract

We revisit the composability of different forms of zero-knowledge proofs when the honest prover strategy is restricted to be polynomial time (given an appropriate auxiliary input). Our results are: 1.) When restricted to efficient provers, the original Goldwasser--Micali--Rackoff (GMR) definition of zero knowledge (STOC `85), here called plain zero knowledge, is closed under a constant number of sequential compositions (on the same input). This contrasts with the case of unbounded provers, where Goldreich and Krawczyk (ICALP `90, SICOMP `96) exhibited a protocol that is zero knowledge under the GMR definition, but for which the sequential composition of 2 copies is not zero knowledge. 2.) If we relax the GMR definition to only require that the simulation is indistinguishable from the verifier's view by uniform polynomial-time distinguishers, with no auxiliary input beyond the statement being proven, then again zero knowledge is not closed under sequential composition of 2 copies. 3.) We show that auxiliary-input zero knowledge with efficient provers is not closed under parallel composition of 2 copies under the assumption that there is a secure key agreement protocol (in which it is easy to recognize valid transcripts). Feige and Shamir (STOC `90) gave similar results under the seemingly incomparable assumptions that (a) the discrete logarithm problem is hard, or (b) \(UP \nsubseteq BPP \) and one-way functions exist.