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The (True) Complexity of Statistical Zero Knowledge (Extended Abstract)
 Proceedings of the 22nd Annual ACM Symposium on the Theory of Computing, ACM
, 1990
"... ) Mihir Bellare Silvio Micali y Rafail Ostrovsky z MIT Laboratory for Computer Science 545 Technology Square Cambridge, MA 02139 Abstract Statistical zeroknowledge is a very strong privacy constraint which is not dependent on computational limitations. In this paper we show that given a comp ..."
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Cited by 44 (19 self)
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) Mihir Bellare Silvio Micali y Rafail Ostrovsky z MIT Laboratory for Computer Science 545 Technology Square Cambridge, MA 02139 Abstract Statistical zeroknowledge is a very strong privacy constraint which is not dependent on computational limitations. In this paper we show that given a complexity assumption a much weaker condition suffices to attain statistical zeroknowledge. As a result we are able to simplify statistical zeroknowledge and to better characterize, on many counts, the class of languages that possess statistical zeroknowledge proofs. 1 Introduction An interactive proof involves two parties, a prover and a verifier, who talk back and forth. The prover, who is computationally unbounded, tries to convince the probabilistic polynomial time verifier that a given theorem is true. A zeroknowledge proof is an interactive proof with an additional privacy constraint: the verifier does not learn why the theorem is true [11]. That is, whatever the polynomialtime verif...
OneWay Functions, Hard on Average Problems, and Statistical ZeroKnowledge Proofs (Extended Abstract)
 IN PROCEEDINGS OF THE 6TH ANNUAL STRUCTURE IN COMPLEXITY THEORY CONFERENCE
, 1991
"... In this paper, we study connections among oneway functions, hard on the average problems, and statistical zeroknowledge proofs. In particular, we show how these three notions are related and how the third notion can be better characterized, assuming the first one. ..."
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In this paper, we study connections among oneway functions, hard on the average problems, and statistical zeroknowledge proofs. In particular, we show how these three notions are related and how the third notion can be better characterized, assuming the first one.
On the Knowledge Complexity of ...
 In 37th FOCS
, 1996
"... We show that if a language has an interactive proof of logarithmic statistical knowledgecomplexity, then it belongs to the class AM \ co AM. Thus, if the polynomial time hierarchy does not collapse, then NPcomplete languages do not have logarithmic knowledge complexity. Prior to this work, ther ..."
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Cited by 27 (7 self)
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We show that if a language has an interactive proof of logarithmic statistical knowledgecomplexity, then it belongs to the class AM \ co AM. Thus, if the polynomial time hierarchy does not collapse, then NPcomplete languages do not have logarithmic knowledge complexity. Prior to this work, there was no indication that would contradict NP languages being proven with even one bit of knowledge. Our result is a common generalization of two previous results: The rst asserts that statistical zero knowledge is contained in AM \ co AM [F89, AH91], while the second asserts that the languages recognizable in logarithmic statistical knowledge complexity are in BPP NP [GOP94]. Next, we consider the relation between the error probability and the knowledge complexity of an interactive proof. Note that reducing the error probability via repetition is not free: it may increase the knowledge complexity. We show that if the negligible error probability (n) is less than 2 3k(n) (where k(n) is the knowledge complexity) then the language proven is in the third level of the polynomial time hierarchy (specically, it is in AM NP . In the standard setting of negligible error probability, there exist PSPACEcomplete languages which have sublinear knowledge complexity. However, if we insist, for example, that the error probability is less than 2 n 2 , then PSPACEcomplete languages do not have subquadratic knowledge complexity, unless PSPACE= P 3 . In order to prove our main result, we develop an AM protocol for checking that a samplable distribution D has a given entropy h. For any fractions ; , the verier runs in time polynomial in 1= and log(1=) and fails with probability at most to detect an additive error in the entropy. We believe that this ...
Making ZeroKnowledge Provers Efficient
 Proceedings of the 24th Annual Symposium on the Theory of Computing, ACM
, 1995
"... We look at the question of how powerful a prover must be to give a zeroknowledge proof. We present the first unconditional bounds on the complexity of a statistical ZK prover. The result is that if a language possesses a statistical zeroknowledge then it also possesses a statistical zeroknowledge ..."
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We look at the question of how powerful a prover must be to give a zeroknowledge proof. We present the first unconditional bounds on the complexity of a statistical ZK prover. The result is that if a language possesses a statistical zeroknowledge then it also possesses a statistical zeroknowledge proof in which the prover runs in probabilistic, polynomial time with an NP oracle. Previously this was only known given the existence of oneway permutations. Extending these techniques to protocols of knowledge complexity k(n) ? 0, we derive bounds on the time complexity of languages of "small" knowledge complexity. Underlying these results is a technique for efficiently generating an "almost" random element of a set S 2 NP. Specifically, we construct a probabilistic machine with an NP oracle which, on input 1 n and ffi ? 0 runs in time polynomial in n and lg ffi \Gamma1 , and outputs a random string from a distribution within distance ffi of the uniform distribution on S " f0; 1g n ...
Computational Complexity and Knowledge Complexity
 In Proc. 26th STOC
, 1996
"... We study the computational complexity of languages which have interactive proofs of logarithmic knowledgecomplexity. We show that all such languages can be recognized in BPP NP . Prior to this work, for languages with greaterthanzero knowledgecomplexity only trivial computational complexity bo ..."
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Cited by 5 (1 self)
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We study the computational complexity of languages which have interactive proofs of logarithmic knowledgecomplexity. We show that all such languages can be recognized in BPP NP . Prior to this work, for languages with greaterthanzero knowledgecomplexity only trivial computational complexity bounds were known. In the course of our proof, we relate statistical knowledgecomplexity with perfect knowledgecomplexity; specifically, we show that, for the honest verifier, these hierarchies coincide, up to a logarithmic additive term. An extended abstract of this paper appeared in the 26th ACM Symposium on Theory of Computing (STOC 94), held in Montreal, Quebec, Canada, May 2325, 1994. y Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel. Email: oded@wisdom.weizmann.ac.il. Supported by grant no. 9200226 from the United States  Israel Binational Science Foundation, Jerusalem, Israel. z Bell Communications Research, 445 South ...
Abstract The (True) Complexity of Statistical Zero Knowledge (Extended Abstract)
"... Statistical zeroknowledge is a very strong privacy constraint which is not dependent on computational limitations. In this paper we showthatgiven a complexity assumption a much weaker condition su ces to attain statistical zeroknowledge. As a result we are able to simplify statistical zeroknowled ..."
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Statistical zeroknowledge is a very strong privacy constraint which is not dependent on computational limitations. In this paper we showthatgiven a complexity assumption a much weaker condition su ces to attain statistical zeroknowledge. As a result we are able to simplify statistical zeroknowledge and to better characterize, on many counts, the class of languages that possess statistical zeroknowledge proofs. 1
Making ZeroKnowledge Provers Efficient
, 1995
"... Abstract We look at the question of how powerful a prover must be to give a zeroknowledge proof. We present the first unconditional bounds on the complexity of a statistical ZK prover. The result is that if a language possesses a statistical zeroknowledge then it also possesses a statistical zero ..."
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Abstract We look at the question of how powerful a prover must be to give a zeroknowledge proof. We present the first unconditional bounds on the complexity of a statistical ZK prover. The result is that if a language possesses a statistical zeroknowledge then it also possesses a statistical zeroknowledge proof in which the prover runs in probabilistic, polynomial time with an NP oracle. Previously this was only known given the existence of oneway permutations. Extending these techniques to protocols of knowledge complexity k(n) ? 0, we derive bounds on the time complexity of languages of &quot;small &quot; knowledge complexity. Underlying these results is a technique for efficiently generating an &quot;almost &quot; random element of a set S 2 N P. Specifically, we construct a probabilistic machine with an NP oracle which, on input 1n and ffi? 0 runs in time polynomial in n and lg ffi \Gamma 1, and outputs a random string from a distribution within distance ffi of the uniform distribution on S &quot; f0; 1gn.
Knowledge Complexity Versus Computational Complexity And The Hardness Of Approximations
, 1995
"... : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 List of Symbols and Abbreviations 3 1 Introduction 4 1.1 Knowledge Complexity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 1.1.1 The Realtion of Statistical to Perfect Knowledge Complexity : : : : ..."
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: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 List of Symbols and Abbreviations 3 1 Introduction 4 1.1 Knowledge Complexity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 1.1.1 The Realtion of Statistical to Perfect Knowledge Complexity : : : : : 6 1.1.2 Efficient Almost Uniform Generation : : : : : : : : : : : : : : : : : : 7 1.1.3 Motivation for studying KC : : : : : : : : : : : : : : : : : : : : : : : 8 1.1.4 Remarks : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 1.2 The Hardness of Approximations : : : : : : : : : : : : : : : : : : : : : : : : 9 1.2.1 Two categories of optimization problems : : : : : : : : : : : : : : : : 11 1.2.2 The gaplocation parameter : : : : : : : : : : : : : : : : : : : : : : : 12 1.2.3 Summary of results : : : : : : : : : : : : : : : : : : : : : : : : : : : : 14 1.2.4 Summary of the motivation for proving hardness at gaplocation 1 : : 15 1.2.5 Related results : : : : : : : : : :...