Low Communication 2-Prover Zero-Knowledge Proofs for NP

We exhibit a two-prover perfect zero-knowledge proof system for 3-SAT. In this protocol, the verifier asks a single message to each prover, whose size grows logarithmically in the size of the 3-SAT formula. Each prover's answer consists of only a constant number of bits. The verifier will always accept correct proofs. Given an unsatisfiable formula S the verifier will reject with probability at least ?((|S| - max-sat(S))/|S|, where max-sat(S) denotes the maximum number of clauses of S that may be simultaneously satisfied, and |S| denotes the total number of clauses of S. Using a recent result by Arora et al [2], we can construct for any language in NP a protocol with the property that any nonmember of the language be rejected with constant probability.

[1]  Joe Kilian,et al.  Uses of randomness in algorithms and protocols , 1990 .

[2]  Carsten Lund,et al.  Nondeterministic exponential time has two-prover interactive protocols , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[3]  L. Fortnow,et al.  On the power of multi-power interactive protocols , 1988, [1988] Proceedings. Structure in Complexity Theory Third Annual Conference.

[4]  Carsten Lund,et al.  Proof verification and hardness of approximation problems , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[5]  László Lovász,et al.  Approximating clique is almost NP-complete , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[6]  Adi Shamir,et al.  A One-Round, Two-Prover, Zero-Knowledge Protocol for NP , 1991, CRYPTO.

[7]  David A. Mix Barrington,et al.  Bounded-width polynomial-size branching programs recognize exactly those languages in NC1 , 1986, STOC '86.

[8]  Sanjeev Arora,et al.  Probabilistic checking of proofs; a new characterization of NP , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[9]  Joe Kilian,et al.  A note on efficient zero-knowledge proofs and arguments (extended abstract) , 1992, STOC '92.

[10]  Leonid A. Levin,et al.  Checking computations in polylogarithmic time , 1991, STOC '91.

[11]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[12]  Uriel Feige.  On the success probability of the two provers in one-round proof systems , 1991, [1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference.

[13]  Richard J. Lipton,et al.  PSPACE is provable by two provers in one round , 1991, [1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference.

[14]  Richard J. Lipton,et al.  On bounded round multiprover interactive proof systems , 1990, Proceedings Fifth Annual Structure in Complexity Theory Conference.

[15]  Avi Wigderson,et al.  Multi-prover interactive proofs: how to remove intractability assumptions , 2019, STOC '88.

[16]  Adi Shamir,et al.  Fully parallelized multi prover protocols for NEXP-time , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[17]  László Lovász,et al.  Two-prover one-round proof systems: their power and their problems (extended abstract) , 1992, STOC '92.

[18]  Richard J. Lipton,et al.  Playing Games of Incomplete Information , 1990, Symposium on Theoretical Aspects of Computer Science.