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A Taxonomy of Proof Systems
 BASIC RESEARCH IN COMPUTER SCIENCE, CENTER OF THE DANISH NATIONAL RESEARCH FOUNDATION
, 1997
"... Several alternative formulations of the concept of an efficient proof system are nowadays coexisting in our field. These systems include the classical formulation of NP , interactive proof systems (giving rise to the class IP), computationallysound proof systems, and probabilistically checkable pro ..."
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Several alternative formulations of the concept of an efficient proof system are nowadays coexisting in our field. These systems include the classical formulation of NP , interactive proof systems (giving rise to the class IP), computationallysound proof systems, and probabilistically checkable proofs (PCP), which are closely related to multiprover interactive proofs (MIP). Although these notions are sometimes introduced using the same generic phrases, they are actually very different in motivation, applications and expressive power. The main objective of this essay is to try to clarify these differences.
Probabilistic Proof Systems  A Survey
 IN SYMPOSIUM ON THEORETICAL ASPECTS OF COMPUTER SCIENCE
, 1996
"... Various types of probabilistic proof systems have played a central role in the development of computer science in the last decade. In this exposition, we concentrate on three such proof systems  interactive proofs, zeroknowledge proofs, and probabilistic checkable proofs  stressing the essen ..."
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Cited by 5 (0 self)
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Various types of probabilistic proof systems have played a central role in the development of computer science in the last decade. In this exposition, we concentrate on three such proof systems  interactive proofs, zeroknowledge proofs, and probabilistic checkable proofs  stressing the essential role of randomness in each of them.
Making classical honest verifier zero knowledge protocols secure against quantum attacks
 In 35th International Colloquium on Automata, Languages and Programming (ICALP), volume 5126 of Lecture Notes in Computer Science
, 2008
"... We show that any problem that has a classical zeroknowledge protocol against the honest verifier also has, under a reasonable condition, a classical zeroknowledge protocol which is secure against all, possibly cheating classical and quantum polynomial time verifiers. Here we refer to the generaliz ..."
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Cited by 4 (0 self)
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We show that any problem that has a classical zeroknowledge protocol against the honest verifier also has, under a reasonable condition, a classical zeroknowledge protocol which is secure against all, possibly cheating classical and quantum polynomial time verifiers. Here we refer to the generalized notion of zeroknowledge with classical and quantum auxiliary inputs respectively. Our condition on the original protocol is that, for positive instances of the problem, the simulated message transcript should be quantum computationally indistinguishable from the actual message transcript. This is a natural strengthening of the notion of honest verifier computational zeroknowledge, and includes in particular, the complexity class of honest verifier statistical zeroknowledge. Our result answers an open question of Watrous [Wat06], and generalizes classical results by Goldreich, Sahai and Vadhan [GSV98], and Vadhan [Vad06] who showed that honest verifier statistical, respectively computational, One of the main impacts of quantum computation thus far has been its potential implications for cryptography. Public key cryptography, a central concept in cryptography, is used to protect web transactions, and its security relies on the hardness of certain number theory problems. Exponential speedups by quantum computers
Texts in Computational Complexity: Probabilistic Proof Systems
, 2006
"... A proof is whatever convinces me. Shimon Even (19352004) Various types of probabilistic proof systems have played a central role in the development of computer science in the last couple of decades. In this text, we concentrate on three such proof systems: interactive proofs, zeroknowledge proofs, ..."
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A proof is whatever convinces me. Shimon Even (19352004) Various types of probabilistic proof systems have played a central role in the development of computer science in the last couple of decades. In this text, we concentrate on three such proof systems: interactive proofs, zeroknowledge proofs, and probabilistic checkable proofs. These proof systems share a common (untraditional) feature they carry a probability of error; yet, this probability is explicitly bounded and can be reduced by successive application of the proof system. The gain in allowing this untraditional relaxation is substantial, as demonstrated by the three results mentioned in the summary. Summary: The association of efficient procedures with deterministic polynomialtime procedures is the basis for viewing NPproof systems as the canonical formulation of proof systems (with efficient verification procedures). Allowing probabilistic verification procedures and, moreover, ruling by statistical evidence gives rise to various types of probabilistic proof systems. These probabilistic proof systems carry a probability of error (which is explicitly bounded), but they offer various advantages over the traditional (deterministic and errorless) proof systems.
Interactive and probabilistic proofchecking
 Annals of Pure and Applied Logic
, 2000
"... The notion of efficient proofchecking has always been central to complexity theory, and it gave rise to the definition of the class NP. In the last 15 years there has been a number of exciting, unexpected and deep developments in complexity theory that exploited the notion of randomized and interac ..."
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The notion of efficient proofchecking has always been central to complexity theory, and it gave rise to the definition of the class NP. In the last 15 years there has been a number of exciting, unexpected and deep developments in complexity theory that exploited the notion of randomized and interactive proofchecking. Results developed along this line of research have diverse and powerful applications in complexity theory, cryptography, and the theory of approximation algorithms for combinatorial optimization problems. In this paper we survey the main lines of developments in interactive and probabilistic proofchecking, with an emphasis on open questions.