A. Shamir, "IP = PSPACE", in Proceedings, 31st Annual IEEE Symposium on Foundations of Computer Science, pages 11-15, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Interactive Proof Systems of Goldwasser et al. 24] are also among examples of games in which one player plays randomly whereas the other existentially picks a strategy. Goldwasser and Sipser [25, 26] have proved the equivalence of Interactive Proof Systems and Arthur Merlin games. Shamir [27] proves both problems are in the same complexity class (PSPACE complete) Another special class of games is Solitaire games. Solitaire games restrict the play of one of the players to be completely deterministic after the player s first move. These games are investigated by Ladner and Norman ....

A. Shamir, "IP = PSPACE", in Proceedings, 31st Annual IEEE Symposium on Foundations of Computer Science, pages 11-15, 1990.

....] Just as in the case when L 2 NP , when L 2 IP , membership in L is efficiently verifiable since the verifier runs in polynomial time and determines membership correctly with probability very close to one. However, IP is thought to strictly contain NP since it has recently been shown by Shamir [S] that IP = PSPACE. In addition to defining interactive proofs, Goldwasser, Micali, and Rackoff [GMR] further defined zero knowledge interactive proofs. The zero knowledge definition was motivated by cryptographic considerations (see for example, GMR2] O] GMW] Informally, a prover is ....

Shamir, Adi, "IP = PSPACE," manuscript.

....Theorem 4.2 [Ostrovsky Wigderson [OW93] Go98, Theorem 4.5.4] Assume that there exists a ZK language outside ETA. Then there exist uniform one way functions. 6 Furthermore, it is well known that IP=ZK=PSPACE assuming the existence of non uniform one way functions [ImYu87] BGG 88][Sh92] 18 Now we derive the relationship between uniform one wayness and restricted uniform correlation intractability by combining Theorem 3.9 and 4.2. Theorem 4.3 Proving the implication, if uniform one way functions exist then restricted uniform correlation intractable function ensembles exist , ....

A. Shamir, "IP=PSPACE, " Journal of ACM, Vol. 39, No. 4, pp. 869-877, 1992.

....that there must be a gap in acceptance probabilities: If x 2 L, then V must accept with probability at least 2=3, and, if x 62 L, then V must accept with probability at most 1=3. One of the most highly acclaimed results in computational complexity theory, proved by Lund et al. 14] and Shamir [19], is that the (seemingly very stringent) requirement of this (1=3; 2=3) gap does not change the class of languages accepted: poly(n) round Arthur Merlin Games also recognize exactly PSPACE. 3 Probabilistically Checkable Debate Systems In the Alternating Polynomial Time, Games Against Nature, and ....

....at, and yet the referee still decides the winner correctly. In order to encode games to permit such efficient refereeing, Condon et al. 6, 7] exploit and extend the probabilistically checkable coding techniques developed in the PCP characterization of NP [1, 20] In conclusion, the results of [5, 6, 7, 14, 17, 19] demonstrate that the identification of PSPACE with zero sum, perfection information, polynomial depth games is extremely robust. Numerous variations on the computational model of a game between two strategic players that is judged after it is played by a polynomialtime referee have been studied, ....

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A. Shamir, "IP = PSPACE," Journal of the Association for Computing Machinery, 39 (1992), pp. 869-877.

....see all of the communication between the referee and the other player) or imperfect recall (does not remember all the information that he himself once knew) This framework allows us to 1. Unify and generalize the game theoretic aspects of ealier work on the complexity of interactive computation [BFL91, CKS81, FRS94, FST88, LFKN92, Sh91]. For example, game theory is a natural framework in which to state the equivalence of oracle proof systems and multiprover proof systems given in [FRS94] because oracles and teams of noncommunicating provers are both examples of players with imperfect recall. An oracle has imperfect recall ....

A. Shamir, "IP = PSPACE," Journal of the ACM, 39 (1992), pp. 869--877.

....shown that all problems in NP are interactively provable using a diploma of O(lg n) bits (Arora and Safra, 1992) An instance of a problem is verifiable in IP iff it is computable in PSpace. P NP PSpace NEXPTime Figure 4: Possible Complexity Hierarchy Theorem 1 (Shamir, 1992) IP = PSpace The class of problems that can be interactively proved using two provers and one verifier is MIP , where the two provers are arbitrarily bounded machines (1 CMs) Technically, these two provers don t communicate with each other, since the verifier plays them off each other. For ....

Shamir, A. (1992) "IP = PSPACE," Journal of the ACM, Vol. 39, No. 4, 869-877.

....of super polynomial time machines. For example, we are able to say in what sense the which handles the entire sequence of formulae. On the other hand, one cannot ignore the case in which something is sent by Alice since this case is not negligible. interactive proofs introduced by Shamir [17], in order to demonstrate that IP=PSPACE, constitute proofs of knowledge. Most proofs of knowledge (e.g. the proof of knowledge of an isomorphism used by [12] see Appendix E) are constructed by iterating some atomic protocol. Typically, these atomic protocols have the property that one can ....

A. Shamir, "IP=PSPACE," Proceedings of the 31st Symposium on Foundations of Computer Science, IEEE, 1990, pp. 11-15.

....a quantum computer cannot invert one way functions, but only proves this for one way oracles, i.e. black box functions given as a subroutine which the quantum computer is not allowed to look inside. Such oracle results have been misleading in the past, most notably in the case of IP = PSPACE [15, 28]. A third approach, which we take, is to solve in BQP some well studied problem for which no polynomial time algorithm is known. This shows that the extra power conferred by quantum interference is at least hard to achieve using classical computation. Both Bernstein and Vazirani [5] and Simon [29] ....

A. Shamir, "IP = PSPACE," in Proc. 31th Ann. Symp. on Foundations of Computer Science, pp. 11--15, IEEE Computer Society Press (1990).

....message, and is allowed to make errors with probability bounded away from 1 2 . The probabilistic relaxation of the verifier s decision is the only additional power that this proof system has over the NP proof system. to the class I IP defined in [19] and recently shown to equal I PSPACE [26]) However, we are not interested here in the complexity theoretic aspects of interactive proofs but rather in their utility in practice. Remark 20 : Definition 9 has some non uniform flavour; namely, the quantification over all x s rather than requiring that it is infeasible to find x s for ....

Shamir, A., "IP = PSPACE", Proc. 31st FOCS, pp. 11--15, 1990.

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Shamir, A., "IP = PSPACE", proceedings of the 31 st IEEE Symposium on Foundations of Computer Science, 1990.

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A. Shamir. "IP=PSPACE." J. ACM, 39 (1992), 869--877.

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Shamir, A., "IP = PSPACE", Proc. 31st IEEE Symp. on Foundations of Computer Science, 1990, pp. 11-15.

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