On Concurrent and Resettable Zero-Knowledge Proofs for NP

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A proof is concurrent zero-knowledge if it remains zero-knowledge when many copies of the proof are run in an asynchronous environment, such as the Internet. It is known that zero-knowledge is not necessarily preserved in such an environment. Designing concurrent zero-knowledge proofs is a fundamental issue in the study of zero-knowledge since known zero-knowledge protocols cannot be run in a realistic modern computing environment. In this paper we present a concurrent zero-knowledge proof systems for all languages in NP. Currently, the proof system we present is the only known proof system that retains the zero-knowledge property when copies of the proof are allowed to run in an asynchronous environment. Our proof system has O(² k) rounds (for a security parameter k), which is almost optimal, as it is shown by Canetti Kilian Petrank and Rosen that black-box concurrent zero-knowledge requires Ω(k) rounds. Canetti, Goldreich, Goldwasser and Micali introduced the notion of resettable zero-knowledge, and modified an earlier version of our proof system to obtain the first resettable zero-knowledge proof system. This protocol requires kθ⁽¹⁾ rounds. We note that their technique also applies to our current proof system, yielding a resettable zero-knowledge proof for NP with O(² k) rounds.