M. Ben-Or, S. Goldwasser, J. Kilian and A. Wigderson. Multi-Prover interactive proofs: How to remove intractability assumptions. STOC 88.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....bits and proofs of size l. A language is in FPCP[ r; f ] if for every function k(n) 1) there is a function k (n) o(k(n) such that L is in FPCP[ k k )r; k k )f; 2 k ; 2 r ] We ll also discuss veri ers who talk to a collection of p = p(n) provers rather than to a proof string [8]. De nitions for such things are standard. If is a probability space then R denotes the operation of selecting an element at random according to and Pr R [ is the corresponding probability. If S is a set then R S is the operation of selecting uniformly at random from S. ....

M. Ben-Or, S. Goldwasser, J. Kilian and A. Wigderson. Multi-Prover interactive proofs: How to remove intractability assumptions. STOC 88.

....the last two decades. The best known ones are interactive proofs [20] zeroknowledge proofs [20] and probabilistic checkable proofs [17, 5, 12, 2] but other notions such as various types of computationally sound proofs (e.g. arguments [10] and CS proofs [23] and multiprover interactive proofs [9] have made a prominent appearance as well. Do we really need yet another type of probabilistic proof systems We believe that the answer is positive. The number of di#erent related notions we need is exactly the number of di#erent notions that are natural, interesting and or useful. Confining ....

M. Ben-Or, S. Goldwasser, J. Kilian and A. Wigderson. Multi-Prover Interactive Proofs: How to Remove Intractability. In 20th STOC, pages 113--131, 1988.

....the normalized Hamming distance and 0 is an absolute constant. Good encoding functions are known to exist (cf. MS, Chapter 10.11] 3 De nition of encoding schemes We recall the de nition of the single round multi prover interactive proof systems of Ben Or, Goldwasser, Kilian, and Wigderson [BGKW]. We use the notation of [BGLR] A MIP consists of p provers and a veri er. Each prover is a function from questions to answers: P i : q a where q is the question size and a the answer size. The veri er is a polynomial time machine receiving the input x and a random string from r . ....

M. Ben-Or, S. Goldwasser, J. Kilian, and A. Wigderson. Multi-prover interactive proofs: how to remove intractability assumptions. In: Proceedings of the 20th Annual ACM Symposium on Theory of Computing, 1988, 113-131.

....to convince the verifier that some assertion is true. The model of the interactive proofs evolved over time, partly motivated by efforts to understand the model better. One such model was that of multi prover interactive proof systems (MIP) introduced by Ben Or, Goldwasser, Kilian and Wigderson [12]. In this model, a single verifier interacts with multiple provers to verify a given assertion. The MIP proof systems influenced the development of PCPs in two significant ways. On the one hand, many technical results about PCPs go through MIP proof systems, in essential ways. More important to ....

Michael Ben-Or, Shafi Goldwasser, Joe Kilian, and Avi Wigderson. Multiprover interactive proofs: How to remove intractability assumptions. In Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing, pages 113-131, Chicago, Illinois, 2-4 May 1988.

.... needs to be decrypted jointly by a group of users, instead of by a single user, was addressed in [17] and later in [26] The first solution presented is not very secure and the second one is far from practical since it relies on mental games (general secure distributed computation) mechanisms [43, 4, 13]. 3 Basic schemes In this section, for simplicity, we mainly follow the description used in [19] and discuss a generalization of it in Section 4.4. A cryptosystem corresponds to evaluate a function with two inputs. One of them is the key, and the other one varies from application to ....

....cardinality larger or equal to t, they will not be able to perform the threshold computation in (5) We refer the reader to [20, 1, 6, 49] 4. 4 Generalizations g is not homomorphic What if the function g is not homomorphic This problem in its generality corresponds with the mental games problem [43, 4, 13]. In its generality no practical solution has been proposed to address this problem. For some algorithms, such as DSS, a practical approach may be desirable. This was studied in [50, 42] It should be noted that, even for the non robust schemes, there is a significant difference between those ....

M. Ben-Or, S. Goldwasser, J. Kilian, and A. Wigderson. Multi-prover interactive proofs: How to remove intractability assumptions. In Proceedings of the twentieth annual ACM Symp. Theory of Computing, STOC, pp. 113--131, May 2--4, 1988.

.... In the last 15 years, this study has proved to be exceedingly successful in complexity theory [ZH86, BHZ87, ZF87, LFKN90, Sh90] Eventually the study of these proof systems (and multi prover systems) led to perhaps the most spectacular achievement in the Theory of Computing in the last decade [BGKW88, LFKN90, Sh90, BFL91, BFLS91, FGLSM91, AS92, ALMSS92]. It is well known that some traditional complexity classes can be characterized by operators. For example NP = 9 P; 2 = 9 8 P, using the existential and universal operators 9 and 8 respectively. They have been used fruitfully to prove (or to give simpler proofs of known) relations ....

M. Ben-Or, S. Goldwasser, J. Kilian, and A. Wigderson, Multiprover interactive proofs: How to remove the intractability assumptions, in Proc. 20th Ann. ACM Symp. Theory of Computing, 1988, 113-131.

....result, due to Lund, Fortnow, Karlo and Nisan [77] and Shamir [94] has shown that every language in PSPACE which is suspected to be a much larger class than NP admits such interactive membership proofs. Another variant of proof veri cation, due to Ben Or, Goldwasser, Kilian and Wigderson [24], involves a probabilistic polynomialtime veri er interacting with more than one mutually non interacting provers. The class of languages with such interactive proofs is called MIP (for Multi prover Interactive Proofs) Fortnow, Rompel and Sipser [46] gave an equivalent de nition of MIP as ....

M. Ben-Or, S. Goldwasser, J. Kilian, and A. Wigderson. Multi-prover interactive proofs: How to remove intractability assumptions. Proceedings of the Twentieth Annual Symposium on the Theory of Computing, ACM, 1988.

....the last two decades. The best known ones are interactive proofs [20] zeroknowledge proofs [20] and probabilistic checkable proofs [17, 5, 12, 2] but other notions such as various types of computationally sound proofs (e.g. arguments [10] and CS proofs [23] and multiprover interactive proofs [9] have made a prominent appearance as well. Do we really need yet another type of probabilistic proof systems We believe that the answer is positive. The number of different related notions we need is exactly the number of different notions that are natural, interesting and or useful. Confining ....

M. Ben-Or, S. Goldwasser, J. Kilian and A. Wigderson. Multi-Prover Interactive Proofs: How to Remove Intractability. In 20th STOC, pages 113--131, 1988.

....order terms when proving hardness of approximation for set cover. 2 A multiprover proof system Our result is based on a reduction from a multiprover proof system. In this section we explain the construction of the proof system. Readers not familier with multiprover proof systems are referred to [4] (or other references in our paper) Consider the problem of MAX 3SAT 5. Input: A CNF formula with n variables and 5n=3 clauses, in which every clause contains exactly three literals (a literal is a Boolean variable in either positive or negated form) every variable appears in exactly five ....

M. Ben-Or, S. Goldwasser, J. Kilian, A. Wigderson. "Multi-prover interactive proofs: how to remove intractability assumptions". Proc. 20th STOC, 1988, 113--131.

....that ja n j q(n) and L =n = fx 2 Sigma n j hx; a n i 2 Bg. It is well known that P=poly coincides with the class of languages which can be recognized by a non uniform family of polynomial size circuits. The definition of multiprover interactive proof systems first appeared in Ben Or et al. [BGKW88] and Babai et al. BFL91] Definition 2.3 Let p be a polynomial and V be a probabilistic polynomial time machine. There is a multiprover interactive (i.e. MIP) protocol for a language L, if for every n there are provers P 1 ; Delta Delta Delta ; P k , k p(n) such that for every x 2 Sigma ....

M. Ben-Or, S. Goldwasser, J. Kilian, A. Wigderson. Multiprover interactive proofs: How to remove the intractability assumptions. Proc. 20th Ann. ACM Symposium Theory of Computing, 113-131, 1988.

....in a specially coded form, then the dependenceon n can be dropped. However, encoding the input requires Omega Gamma n) time. 2 2 The model Interactive proof systems (IPS) GMR89] and probabilistically checkable proof systems (PCPS) FRS94] equivalent in power to multiple prover proof systems [BGKW88], see also [FGL 96, AS98, BFLS90] and to function restricted IP [FRS94] can be used to convince a polynomial time verifier of the correctness of a decision problem computation. Definitions of IP which parametrize the runtime of the verifier appear in [Con91, FL93] CS proofs [Mic94] extend the ....

M. Ben-Or, S. Goldwasser, J. Kilian, and A. Wigderson. Multi-prover interactive proofs: How to remove intractability assumptions. Proc. 20th Symposium on Theory of Computing, pp. 113--131, 1988.

.... In the last 15 years, this study has proved to be exceedingly successful in complexity theory [ZH86, BHZ87, ZF87, LFKN90, Sh90] Eventually the study of these proof systems (and multi prover systems) led to perhaps the most spectacular achievement in the Theory of Computing in the last decade [BGKW88, LFKN90, Sh90, BFL91, BFLS91, FGLSM91, AS92, ALMSS92]. It is well known that some traditional complexity classes can be characterized by operators. For example NP = 9 Delta P; Sigma 2 = 9 Delta 8 Delta P, using the existential and universal operators 9 and 8 respectively. They have been used fruitfully to prove (or to give simpler proofs of ....

M. Ben-Or, S. Goldwasser, J. Kilian, and A. Wigderson, Multiprover interactive proofs: How to remove the intractability assumptions, in Proc. 20th Ann. ACM Symp. Theory of Computing, 1988, 113-131.

....system, the proof consists of p arrays of letters from #. The verifier is only allowed to query 1 letter from each array. Since each letter of # is represented by #log # # bits, the number of bits queried may be viewed as p #log # #. Constructions of such proof systems for NP appeared in [30, 96, 52, 27, 50, 113]. Lund and Yannakakis [108] used these proof systems to prove inapproximability results for SETCOVER and many subgraph maximization problems. The best construction of such proof systems is due to Raz and Safra [114] They show that for each k # # log n, every NP language has a verifier that uses ....

M. Ben-or, S. Goldwasser, J. Kilian, and A. Wigderson. Multi prover interactive proofs: How to remove intractability assumptions. In Proc. 20th ACM Symp. on Theory of Computing, pages 113--121, 1988.

....due to Lund, Fortnow, Karlo# and Nisan [77] and Shamir [94] has shown that every language in PSPACE which is suspected to be a much larger class than NP admits such interactive membership proofs. Another variant of proof verification, due to Ben Or, Goldwasser, Kilian and Wigderson [24], involves a probabilistic polynomial time verifier interacting with more than one mutually non interacting provers. The class of languages with such interactive proofs is called MIP (for Multi prover Interactive Proofs) Fortnow, Rompel and Sipser [46] gave an equivalent definition of MIP as ....

M. Ben-Or, S. Goldwasser, J. Kilian, and A. Wigderson. Multi-prover interactive proofs: How to remove intractability assumptions. Proceedings of the Twentieth Annual Symposium on the Theory of Computing, ACM, 1988.

....in IP (for interactive proofs) if there exists a verifier V that is always convinced when x 2 L, but if x = 2 L then any prover P has only a small probability of convincing V to the contrary. The model of multi prover interactive proofs was introduced by Ben Or, Goldwasser, Kilian, and Wigderson [BGKW88] The model consists of a random polynomial time verifier V communicating with two infinitely powerful provers who cannot communicate with each other during the protocol. The provers try to convince the verifier that the input x is in a language L. A language L is in the class MIP (for ....

M. Ben-or, S. Goldwasser, J. Kilian, and A. Wigderson. Multi prover interactive proofs: How to remove intractability assumptions. In Proc. 20th ACM Symp. on Theory of Computing, pages 113--121, 1988.

....activities. This essay is focused on the definitional and constructive activities mentioned above. Other activities in the foundations of cryptography include the exploration of new directions and the marking of limitations. For example, we mention novel modes of operation such as split entities [16, 45, 96], batching operations [55] off line on line signing [50] and Incremental Cryptography [6, 7] On the limitation side, we mention [84, 66] In particular, 84] indicates that certain tasks (e.g. secret key exchange) are unlikely to be achieved by using a one way function in a black box manner . ....

M. Ben-Or, S. Goldwasser, J. Kilian and A. Wigderson. Multi-Prover Interactive Proofs: How to Remove Intractability. In 20th ACM Symposium on the Theory of Computing, pages 113--131, 1988. 26

....in which the ability to sign messages of a signer is limited to a fixed number k of signatures. It is an identity based signature scheme in which each signature can be used only once. We called such schemes bounded life span . It is based on mental games [SRA81] and it uses zero knowledge tools [Sh85, GMRa89, BGKW88]. Cryptography using one time pad or key is known to be the most secure. There exist similar aimed authentication schemes described in [dWQ90] but their signature adaptations are too complex. There exist also schemes based on non zeroknowledge techniques as described in [Vau93] We combine a ....

M. Ben-Or, S. Goldwasser, J. Killian and A. Wigderson. Multi-prover interactive proofs : How to remove intractability assumptions. Proceedingsof the twentieth annual ACM Symp. Theory of Computing, STOC'88, pp. 113--131, May 2--4,1988.

....This brief summary is focused on the definitional and constructive activities mentioned above. Other activities in the foundations of cryptography include the exploration of new directions and the marking of limitations. For example, we mention novel modes of operation such as split entities [5, 23], batching operations [11] off line on line signing [10] and Incremental Cryptography [1, 2] On the limitation side, we mention [20, 14] In particular, 20] indicates that 1 The only exception to the latter statement is Levin s observation regarding the existence of a universal one way ....

M. Ben-Or, S. Goldwasser, J. Kilian and A. Wigderson. Multi-Prover Interactive Proofs: How to Remove Intractability. In 20th STOC, pages 113--131, 1988.

.... of [FGM 89] from interactive proofs to ones with perfect completeness, and transformations of honest verifier zero knowledge proofs to general zero knowledge proofs [BMO90, OVY93, DGOW95, GSV98] The only exceptions we know of are those that exploit complete problems, such as [GMW91, BGKW88, LFKN92, Sha92] and typically this approach increases complexity to the maximum. For example, another way to prove that every problem possessing an interactive proof also has a public coin interactive proof would be to combine the inclusion IP PSPACE with the direct public coin interactive ....

Michael Ben-Or, Shafi Goldwasser, Joe Kilian, and Avi Wigderson. Multi-prover interactive proofs: How to remove intractability assumptions. In Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing, pages 113--131, Chicago, Illinois, 2--4 May 1988.

....simply refuse to answer a question that would not have been asked by the honest veri er. However, PCPs are formally viewed as sequences of bits; there is no entity in place to judge a question s legitimacy. Consequently, the theorem that everything provable in MIP is provable in zero knowledge [2, 7] does not translate over automatically. Dwork et al. 3] introduce the notion of zeroknowledge or robust PCP s. A robust PCP( V ) is a distribution on PCPs along with a probabilistic polynomial time veri er V . The prover samples a PCP from the given distribution, writes this sample down, ....

M. Ben-Or, S. Goldwasser, J. Kilian, and A. Wigderson. Multi-Prover Interactive Proofs: How to Remove Intractability Assumptions, Proc. of STOC88.

....and approximating the clique problem. In section 4 we improve the efficiency of [5] s proof system and scale it down to complexity classes lower than NEXPTIME. 2 Multi Prover Protocols The model of multi prover interactive proofs was introduced by Ben Or, Goldwasser, Kilian, and Wigderson in [7]. It is defined as follows. 6 Let P 1 ; P 2 be infinitely powerful machines and V be a probabilistic polynomialtime Turing machine, all of which share the same read only input tape. The verifier V shares communication tapes with each P i , but provers P 1 and P 2 have no common tapes except the ....

M. Ben-or, S. Goldwasser, J. Kilian, and A. Wigderson. Multi prover interactive proofs: How to remove intractability. In Proc. 20th ACM Symp. on Theory of Computing, pages 113--131, 1988.

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M. BEN-OR, S. GOLDWASSER, J. KILIAN AND A. WIGDERSON. Multi-Prover interactive proofs: How to remove intractability assumptions. STOC, 1988.

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M. Ben-Or, S. Goldwasser, J. Kilian, and A. Wigderson. Multi-prover interactive proofs: How to remove intractability assumptions. Proc. 20th Symposium on Theory of Computing, pp. 113--131, 1988.

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M. Ben-Or, S. Goldwasser, J. Kilian, and A. Wigderson, Multi-prover interactive proofs: How to remove intractability

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Ben-Or, M., Goldwasser, S., Kilian, J., and Wigderson, A., "Multi-Prover Interactive Proofs: How to Remove Intractability", proceedings of the 20 th ACM Symposium on Theory of Computing, pp. 113-131, 1988.

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