Precise zero-knowledge, introduced by Micali and Pass [STOC'06], captures the idea that a view of any verifier can be indifferently reconstructed. Though there are some constructions of precise zero-knowledge, constant-round constructions are unknown to exist. This paper is towards constant-round constructions of precise zero-knowledge. The results of this paper are as follows. We propose a relaxation of precise zero-knowledge that captures the idea that with a probability arbitrarily polynomially close to 1 a view of any verifier can be indifferently reconstructed, i.e., there exists a simulator (without having q(n),p(n,t) as input) such that for any polynomial q(n), there is a polynomial p(n,t) satisfying with probability at least $1-rac{1},{q(n)},$, the view of any verifier in every interaction can be reconstructed in p(n,T) time by the simulator whenever the verifier's running-time on this view is T. Then we show the impossibility of constructing constant-round protocols satisfying our relaxed definition with all the known techniques. We present a constant-round precise zero-knowledge argument for any language in NP with respect to our definition, assuming the existence of collision-resistant hash function families (against all n~ (O(loglogn))-size circuits).
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