Martin Furer, Oded Goldreich, Yishay Mansour, Michael Sipser, and Stathis Zachos. On completeness and soundness in interactive proof systems. In Silvio Micali, editor, Advances in Computing Research, volume 5, pages 429--442. JAC Press, Inc., 1989.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....at most Delta (G) ffl Perfect completeness. An interactive proof (P; V ) for L is said to have perfect completeness if the probability of acceptance in the completeness condition is 1. We know that any language L possessing an interactive proof also possesses one with perfect completeness [FGMSZ]. Does any language possessing a competitive interactive proof also possesses a competitive interactive proof with perfect completeness (One of the motivations for this question is the fact that our competitive proof for the special case of quadratic residuosity in Section 6 does not possess ....

M. Furer, O. Goldreich, Y. Mansour, M. Sipser, and S. Zachos. On Completeness and Soundness in Interactive Proof Systems. Advances in Comput. Research Vol. 5, S. Micali ed., JAI Press Inc.

....= yg 2 AM and jf(x)j is polynomially bounded in jxj. To this end, we only have to modify the protocol so that Merlin helps Arthur evaluate the function. A remark about perfect completeness: Finally, one can reduce the rejection probability when the bound is correct to zero by standard techniques [GMS 87] making a one sided error Arthur Merlin game. 4 An overview of the techniques in [AH 91] The main result of this paper is that SKC(O(logn) AM co AM. This generalizes the result of Fortnow [F 89] and Aiello and Hastad [AH 91] stating SZK AM co AM. Let us start by recalling the underlying ....

M. Furer, O. Goldreich, Y. Mansour, M. Sipser and S. Zachos. On Completeness and Soundness in Interactive Proof Systems. Advances in Computing Research: A Research Annual, Vol. 5, Randomness and Computation (ed. S. Micali), 1989, pp. 429-442.

....at most Delta (G) ffl Perfect completeness. An interactive proof (P; V ) for L is said to have perfect completeness if the probability of acceptance in the completeness condition is 1. We know that any language L possessing an interactive proof also possesses one with perfect completeness [FGMSZ]. Does any language possessing a competitive interactive proof also possesses a competitive interactive proof with perfect completeness (One of the motivations for this question is the fact that our competitive proof for the special case of quadratic residuosity in Section 6 does not possess ....

M. Furer, O. Goldreich, Y. Mansour, M. Sipser, and S. Zachos. On Completeness and Soundness in Interactive Proof Systems. Advances in Comput. Research Vol. 5, S. Micali ed., JAI Press Inc.

....00 . Perfect Completeness. 8x 2 L n , Pr(oe R f0; 1g n c ; Proof R Prover(oe; x) Verifier(oe; x; Proof ) 1) 1: In fact, Theorem 3.3 Let L 2 Bounded NIZK. Then L has a Bounded Non Interactive ZKPS with perfect completeness. Proof: Furer, Goldreich, Mansour, Sipser, and Zachos [FuGoMaSiZa] have proved that any AM 2 language has an interactive proof system with perfect completeness. Let now (P; V ) be a Bounded NonInteractive ZKPS for L for which Completeness holds with overwhelming probability. Then modify P as follows. Whenever, the proof generated by P is not accepted by the ....

....R f0; 1g n c ; Proof R Prover(oe; x) Verifier(oe; x; Proof ) 1) 2=3 and 2. 8x 62 L n , for all Turing machines Prover 0 , and for all sufficiently large n, Pr(oe R f0; 1g n c ; Proof R Prover 0 (oe; x) Verifier(oe; x; Proof) 1) 1=3: Moreover, by the result of [FuGoMaSiZa], the proof system (Prover, Verifier) enjoys perfect completeness. Define now the language L 0 = n L 0 (n) where L 0 (n) f(r; x) jrj = n c ; x 2 L n ; and 9w; jwj n c such that Verifier(r; x; w) 1g and L and c are as above. Then x 2 L n iff (r; x) 2 L 0 (n) for most n c ....

M. Furer, O. Goldreich, Y. Mansour, M. Sipser, and S. Zachos, On completeness and Soundness in Interactive Proof Systems, in Advances in Computing Research, vol. 5 Randomness and Computation, S. Micali editor.

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Martin Furer, Oded Goldreich, Yishay Mansour, Michael Sipser, and Stathis Zachos. On completeness and soundness in interactive proof systems. In Silvio Micali, editor, Advances in Computing Research, volume 5, pages 429--442. JAC Press, Inc., 1989.

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M. Furer, O. Goldreich, Y. Mansour, M. Sipser, and S. Zachos. On Completeness and Soundness in Interactive Proof Systems. Advances in Computing Research: a research annual, Vol. 5 (Randomness and Computation, S. Micali, ed.), pp. 429--442, 1989.

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M. Furer, O. Goldreich, Y. Mansour, M. Sipser, and S. Zachos. On Completeness and Soundness in Interactive Proof Systems. Advances in Computing Research: a research annual, Vol. 5 (Randomness and Computation, S. Micali, ed.), pp. 429--442, 1989.

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M. Furer, O. Goldreich, Y. Mansour, M. Sipser, and S. Zachos. On Completeness and Soundness in Interactive Proof Systems. Advances in Computing Research: a research annual, Vol. 5 (Randomness and Computation, S. Micali, ed.), pp. 429--442, 1989.

No context found.

M. Furer, O. Goldreich, Y. Mansour, M. Sipser, and S. Zachos. On Completeness and Soundness in Interactive Proof Systems. Advances in Computing Research: a research annual, Vol. 5 (Randomness and Computation, S. Micali, ed.), pp. 429--442, 1989.

....input z, with probability C(z) As far as polynomial time (or even more powerful) verifiers are concerned any choice of a polynomial time constructible bound, C( which is both non negligibly greater than c( and bounded above by 1 2 PlY( is equivalent. a In fact, following the ideas in [9], one can eliminate the error prob ability in the completeness condition altogether and derive the definition as in the previous section. However, although the last transformation does preserve 8 When saying that these choices are equivalent, as long as the above requirements are satisfied, we ....

M. Furer, O. Goldreich, y. Mansour, M. Sipset, and S. Zachos, "On Completeness and Soundness in Interactive Proof Systems", Advances in Computing Research: a research annual, Vol. 5 (S. Micali, ed.), pp. 429-442, 1989.

....(As we shall see, this does not necessarily hold in the context of zero knowledge proofs. Also, in some sources interactive proofs are defined so that two sided error probability is allowed (rather than requiring perfect completeness as done above) yet, this does not increase their power [44]. Arguments (or Computational Soundness) A fundamental variant on the notion of interactive proofs was introduced by Brassard, Chaum and Cr epeau [21] who relaxed the soundness condition so that it only refers to feasible ways of trying to fool the verifier (rather than to all possible ways) ....

....BPP n coRP require modifying the definition of interactive proofs such that to allow a negligible error also in the completeness condition. Alternatively, zero knowledge proofs for sets in BPP can be constructed by having the prover send a single message that is distributed almost uniformly (cf. [44]) by itself) however, what we seek is zero knowledge proofs for statements that the verifier cannot decide by itself. 4.1 Constructing Zero Knowledge Proofs for NP sets Assuming the existence of commitment schemes , which in turn exist if one way functions exist [76, 68] there exist ....

M. Furer, O. Goldreich, Y. Mansour, M. Sipser, and S. Zachos. On Completeness and Soundness in Interactive Proof Systems. Advances in Computing Research: a research annual, Vol. 5 (Randomness and Computation, S. Micali, ed.), pages 429--442, 1989.

....well known transformations we obtain the claimed result. Specifically, we first reduce the error of the interactive proof by parallel repetition, next transform it into an ArthurMerlin interactive proof [GS] and finally transform it into an Arthur Merlin interactive proof of perfect completeness [FGMSZ]. We stress that all the transformations maintain the number of rounds upto a constant and that the constant round Arthur Merlin hierarchy collapses to one round [Bab] Proof of Proposition 10.8, Parts (3) and (4) For these parts we observe that the proof systems used in the corresponding parts ....

M. F urer, O. Goldreich, Y. Mansour, M. Sipser, and S. Zachos. On completeness and soundness in interactive proof systems. In Advances in Computing Research: a research annual, Vol. 5 (Randomness and Computation, S. Micali, ed.), 1989, pp. 429--442.

.... the verifier uses very little truly new randomness (i.e. there is a session in which the verifier s moves are almost determined by the history of previous sessions) Loosely speaking, the limitations of deterministic verifiers (with respect to interactive proofs and zero knowledge proofs; cf. [16] and [21] respectively) should apply here. 2.3 On the triviality of resettably sound black box zero knowledge In this section we explain why the resettably sound zero knowledge arguments presented in the next subsection are not accompanied (as usual) by a black box simulator. Specifically, we ....

M. Furer, O. Goldreich, Y. Mansour, M. Sipser, and S. Zachos. On Completeness and Soundness in Interactive Proof Systems. Advances in Computing Research: a research annual, Vol. 5 (Randomness and Computation, S. Micali, ed.), pages 429--442, 1989.

.... which the veri er uses very little truly new randomness (i.e. there is a session in which the veri er s moves are almost determined by the history of previous sessions) Loosely speaking, the limitations of deterministic veri ers (with respect to interactive proofs and zero knowledge proofs; cf. [18] and [25] respectively) should apply here. The actual proof is more complex; see below. We start with the main part of the theorem: Suppose that (P; V ) is an interactive proof as in the main part of the theorem, and suppose that on common input x machine V uses a random tape of length m = ....

M. Furer, O. Goldreich, Y. Mansour, M. Sipser, and S. Zachos. On Completeness and Soundness in Interactive Proof Systems. Advances in Computing Research: a research annual, Vol. 5 (Randomness and Computation, S. Micali, ed.), pages 429-442, 1989.

.... the verifier uses very little truly new randomness (i.e. there is a session in which the verifier s moves are almost determined by the history of previous sessions) Loosely speaking, the limitations of deterministic verifiers (with respect to interactive proofs and zero knowledge proofs; cf. [18] and [25] respectively) should apply here. The actual proof is more complex; see below. We start with the main part of the theorem: Suppose that (P; V ) is an interactive proof as in the main part of the theorem, and suppose that on common input x machine V uses a random tape of length m = ....

M. Furer, O. Goldreich, Y. Mansour, M. Sipser, and S. Zachos. On Completeness and Soundness in Interactive Proof Systems. Advances in Computing Research: a research annual, Vol. 5 (Randomness and Computation, S. Micali, ed.), pages 429--442, 1989.

....predicate to the (full) sequence of messages it has sent and received. Public coin proof systems are easier to analyze and manipulate than general interactive proofs, and thus the result of Goldwasser and Sipser [13] by which the former are as powerful as the latter found many applications (e.g. [9, 15, 4]) As mentioned above, the same and more so is true regarding Statistical Zero Knowledge: That is, Okamoto s result [16] i.e. Thm. I) by which public coin HVSZK equals HVSZK, has played a major role in many subsequent results (e.g. his Thm. II as well as in [18, 11] Thus, providing a clear ....

Martin Furer, Oded Goldreich, Yishay Mansour, Michael Sipser, and Stathis Zachos. On completeness and soundness in interactive proof systems. In Silvio Micali, editor, Advances in Computing Research, volume 5, pages 429--442. JAC Press, Inc., 1989.

....into an AM proof system [7] In case of a randomized (smart) reduction, we let the verifier select the random input (to the reduction) and continue as above. This yields a proof system with non perfect completeness, but the (exponentially vanishing) completeness error can be eliminated using [20]. Combining Theorems 3, 4 and 7, we have: Corollary 8 If either GapCVP p n or GapSVP p n is NP hard via smart reductions then coNP AM. It is known that the CVP is NP Hard to approximate within any constant factor, and is hard to approximate within 2 log 1 Gammaffl n unless NP is in e P ....

M. Furer, O. Goldreich, Y. Mansour, M. Sipser, and S. Zachos. On Completeness and Soundness in Interactive Proof Systems. Advances in Computing Research: a research annual, Vol. 5 (Randomness and Computation, S. Micali, ed.), pages 429--442, 1989.

No context found.

M. Furer, O. Goldreich, Y. Mansour, M. Sipser, and S. Zachos. On completeness and soundness in interactive proof systems. In S. Micali, editor, Advances in Computing Research 5: Randomness and Computation, pages 429-442. JAI Press, Greenwich, CT, 1989.

No context found.

M. Furer, O. Goldriech, Y. Mansour, M. Sipser, and S. Zachos. On completeness and soundness in interactive proof systems. S. Micali, editor, Advances in Computing Research 5: Randmness and Computation, pages 429--442, 1989.

No context found.

M. Furer, O. Goldreich, Y. Mansour, M. Sipser, and S. Zachos. On completeness and soundness in interactive proof systems. In S. Micali, editor, Randomness and Computation, volume 5 of Advances in Computing Research, pages 429-442. JAI Press, Greenwich, 1989.

No context found.

Martin Furer, Oded Goldreich, Yishay Mansour, Michael Sipser, and Stathis Zachos. On completeness and soundness in interactive proof systems. In S. Micali, editor, Advances in Computing Research 5: Randomness and Computation, pages 429--442. JAI Press, Greenwich, CT, 1989.

No context found.

M. Furer, O. Goldreich, Y. Mansour, M. Sipser, and S. Zachos. On completeness and soundness in interactive proof systems. Advances in Computing Research: A Research Annual, Randomess and Computation, 5:429--442, 1989.

No context found.

Martin Furer, Oded Goldreich, Yishay Mansour, Michael Sipser, and Stathis Zachos. On completeness and soundness in interactive proof systems. In S. Micali, editor, Advances in Computing Research 5: Randomness and Computation, pages 429--442. JAI Press, Greenwich, CT, 1989.

No context found.

Martin Furer, Oded Goldreich, Yishay Mansour, Michael Sipser, and Stathis Zachos. On completeness and soundness in interactive proof systems. In Silvio Micali, editor, Advances in Computing Research, volume 5, pages 429--442. JAC Press, Inc., 1989.

No context found.

M. Furer, O. Goldreich, Y. Mansour, M. Sipser, and S. Zachos. On completeness and soundness in interactive proof systems. In S. Micali, editor, Randomness and Computation, volume 5 of Advances in Computing Research, pages 429-442. JAI Press, Greenwich, 1989.

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