Round-optimal zero-knowledge proofs of knowledge for NP

It is well known that all the known black-box zero-knowledge proofs of knowledge for NP are nonconstant-round.Whether there exit constant-round black-box zero-knowledge proofs of knowledge for all NP languages under certain standard assumptions is an open problem.This paper focuses on the problem and gives a positive answer by presenting two constructions of constant-round(black-box) zero-knowledge proofs of knowledge for the HC(hamiltonian cycle) problem.By the recent result of Katz,our second construction which relies on the existence of claw-free functions has optimal round complexity(5-round) assuming the polynomial hierarchy does not collapse.