On interactive proofs with a laconic prover

AbstractWe continue the investigation of interactive proofs with bounded communication, as initiated by Goldreich & Håstad (1998). Let L be a language that has an interactive proof in which the prover sends few (say b) bits to the verifier. We prove that the complement $\bar L$ has a constant-round interactive proof of complexity that depends only exponentially on b. This provides the first evidence that for NP-complete languages, we cannot expect interactive provers to be much more “laconic” than the standard NP proof. When the proof system is further restricted (e.g., when b = 1, or when we have perfect completeness), we get significantly better upper bounds on the complexity of $\bar L$.

[1]  Shafi Goldwasser,et al.  Private coins versus public coins in interactive proof systems , 1986, STOC '86.

[2]  Mihir Bellare,et al.  Free Bits, PCPs, and Nonapproximability-Towards Tight Results , 1998, SIAM J. Comput..

[3]  Alan Tucker,et al.  Applied Combinatorics , 1980 .

[4]  Alan Tucker,et al.  Applied combinatorics (3rd ed.) , 1994 .

[5]  Mihir Bellare,et al.  Knowledge on the average—perfect, statistical and logarithmic , 1995, STOC '95.

[6]  David Chaum,et al.  Minimum Disclosure Proofs of Knowledge , 1988, J. Comput. Syst. Sci..

[7]  László Babai,et al.  Trading group theory for randomness , 1985, STOC '85.

[8]  Fred S. Roberts,et al.  Applied Combinatorics , 1984 .

[9]  Peter Bro Miltersen,et al.  Derandomizing Arthur-Merlin Games using Hitting Sets , 1999 .

[10]  V. Arvind,et al.  On Resource-Bounded Measure and Pseudorandomness , 1997, FSTTCS.

[11]  László Babai,et al.  Arthur-Merlin Games: A Randomized Proof System, and a Hierarchy of Complexity Classes , 1988, J. Comput. Syst. Sci..

[12]  Tatsuaki Okamoto.  On Relationships between Statistical Zero-Knowledge Proofs , 2000, J. Comput. Syst. Sci..

[13]  Michael Sipser,et al.  Introduction to the Theory of Computation , 1996, SIGA.

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

[15]  Silvio Micali,et al.  Proofs that yield nothing but their validity or all languages in NP have zero-knowledge proof systems , 1991, JACM.

[16]  Rafail Ostrovsky,et al.  Computational Complexity and Knowledge Complexity , 1994, Electron. Colloquium Comput. Complex..

[17]  Johan Håstad,et al.  Statistical Zero-Knowledge Languages can be Recognized in Two Rounds , 1991, J. Comput. Syst. Sci..

[18]  Lance Fortnow,et al.  The Complexity of Perfect Zero-Knowledge , 1987, Proceeding Structure in Complexity Theory.

[19]  Sanjeev Arora,et al.  Probabilistic checking of proofs: a new characterization of NP , 1998, JACM.

[20]  Gábor Tardos,et al.  On the Knowledge Complexity of NP , 1996, IEEE Annual Symposium on Foundations of Computer Science.

[21]  Oded Goldreich,et al.  Modern Cryptography, Probabilistic Proofs and Pseudorandomness , 1998, Algorithms and Combinatorics.

[22]  Adi Shamir,et al.  IP = PSPACE , 1992, JACM.

[23]  Gábor Tardos,et al.  On the Knowledge Complexity of , 2002, Comb..

[24]  SahaiAmit,et al.  A complete problem for statistical zero knowledge , 2003 .

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

[26]  Amit Sahai,et al.  A complete problem for statistical zero knowledge , 2003, JACM.

[27]  Oded Goldreich,et al.  On the Limits of Nonapproximability of Lattice Problems , 2000, J. Comput. Syst. Sci..

[28]  Tatsuaki Okamoto,et al.  On relationships between statistical zero-knowledge proofs , 1996, STOC '96.

[29]  Amit Sahai,et al.  A complete promise problem for statistical zero-knowledge , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[30]  Carsten Lund,et al.  Algebraic methods for interactive proof systems , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[31]  Silvio Micali,et al.  The knowledge complexity of interactive proof-systems , 1985, STOC '85.

[32]  Oded Goldreich,et al.  Quantifying knowledge complexity , 1999, computational complexity.

[33]  Stathis Zachos,et al.  Does co-NP Have Short Interactive Proofs? , 1987, Inf. Process. Lett..

[34]  Oded Goldreich,et al.  On the Complexity of Interactive Proofs with Bounded Communication , 1998, Inf. Process. Lett..

[35]  Dieter van Melkebeek,et al.  Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses , 1999, STOC '99.