Proofs of membership vs. proofs of knowledge

We investigate the relationship between interactive proofs of membership and interactive proofs of knowledge. Previous results in this area show that many proofs of membership for some languages are also proofs of knowledge of an associated relation, raising the question of whether all proofs of membership are proofs of knowledge. In this paper we clarify the relationship between these two notions of proofs. It turns out that a precise relationship depends on the kind of relation considered. Clearly, any proof of membership is a proof of knowledge for some easy to compute relation. On the other hand, we define a notion of tight relations, referring to relations that capture the computational advantage communicated by a prover to a poly-time verifier in an interactive protocol.

[1]  Manuel Blum,et al.  How to Generate Cryptographically Strong Sequences of Pseudo Random Bits , 1982, FOCS.

[2]  Ran Raz,et al.  A parallel repetition theorem , 1995, STOC '95.

[3]  Rajeev Motwani,et al.  Randomized Algorithms , 1995, SIGA.

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

[5]  Martín Abadi,et al.  On hiding information from an oracle , 1987, STOC '87.

[6]  Martin Tompa,et al.  Random self-reducibility and zero knowledge interactive proofs of possession of information , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[7]  Joan Feigenbaum,et al.  Random-Self-Reducibility of Complete Sets , 1993, SIAM J. Comput..

[8]  Oded Goldreich,et al.  Quantifying knowledge complexity , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[9]  Manuel Blum,et al.  A Simple Unpredictable Pseudo-Random Number Generator , 1986, SIAM J. Comput..

[10]  Michael E. Saks,et al.  Products and Help Bits in Decision Trees , 1999, SIAM J. Comput..

[11]  G. H. C. New.  On coherence , 1989, Nature.

[12]  Manuel Blum,et al.  How to generate cryptographically strong sequences of pseudo random bits , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[13]  Giovanni Di Crescenzo,et al.  On monotone formula closure of SZK , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[14]  Andrew Chi-Chih Yao,et al.  Coherent Functions and Program ( extended abstract ) Checkers , .

[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]  Silvio Micali,et al.  The Knowledge Complexity of Interactive Proof Systems , 1989, SIAM J. Comput..

[17]  Mihir Bellare,et al.  The Complexity of Decision Versus Search , 1991, SIAM J. Comput..

[18]  Martín Abadi,et al.  On Hiding Information from an Oracle , 1987, Proceeding Structure in Complexity Theory.

[19]  Carsten Lund,et al.  Algebraic methods for interactive proof systems , 1992, JACM.

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

[21]  Ran Raz,et al.  A direct product theorem , 1994, Proceedings of IEEE 9th Annual Conference on Structure in Complexity Theory.

[22]  M. Blum,et al.  A simple secure pseudo-random number generator , 1982 .

[23]  Moni Naor,et al.  Amortized Communication Complexity , 1995, SIAM J. Comput..

[24]  Joan Feigenbaum,et al.  On coherence, random-self-reducibility, and self-correction , 1998, computational complexity.

[25]  Amos Fiat,et al.  Zero-knowledge proofs of identity , 1987, Journal of Cryptology.

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

[27]  Moti Yung,et al.  Zero-Knowledge Proofs of Computational Power (Extended Summary) , 1989, EUROCRYPT.

[28]  Mihir Bellare,et al.  On Defining Proofs of Knowledge , 1992, CRYPTO.

[29]  Adi Shamir,et al.  Witness indistinguishable and witness hiding protocols , 1990, STOC '90.

[30]  Joan Feigenbaum,et al.  Lower bounds on random-self-reducibility , 1990, Proceedings Fifth Annual Structure in Complexity Theory Conference.

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