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 non-constant-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.