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58
The knowledge complexity of interactive proof systems
 in Proc. 27th Annual Symposium on Foundations of Computer Science
, 1985
"... Abstract. Usually, a proof of a theorem contains more knowledge than the mere fact that the theorem is true. For instance, to prove that a graph is Hamiltonian it suffices to exhibit a Hamiltonian tour in it; however, this seems to contain more knowledge than the single bit Hamiltonian/nonHamiltoni ..."
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Cited by 1039 (38 self)
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Abstract. Usually, a proof of a theorem contains more knowledge than the mere fact that the theorem is true. For instance, to prove that a graph is Hamiltonian it suffices to exhibit a Hamiltonian tour in it; however, this seems to contain more knowledge than the single bit Hamiltonian/nonHamiltonian. In this paper a computational complexity theory of the "knowledge " contained in a proof is developed. Zeroknowledge proofs are defined as those proofs that convey no additional knowledge other than the correctness of the proposition in question. Examples of zeroknowledge proof systems are given for the languages of quadratic residuosity and quadratic nonresiduosity. These are the first examples of zeroknowledge proofs for languages not known to be efficiently recognizable. Key words, cryptography, zero knowledge, interactive proofs, quadratic residues AMS(MOS) subject classifications. 68Q15, 94A60 1. Introduction. It is often regarded that saying a language L is in NP (that is, acceptable in nondeterministic polynomial time) is equivalent to saying that there is a polynomial time "proof system " for L. The proof system we have in mind is one where on input x, a "prover " creates a string a, and the "verifier " then computes on x and a in time polynomial in the length of the binary representation of x to check that
Entity Authentication and Key Distribution
, 1993
"... Entity authentication and key distribution are central cryptographic problems in distributed computing  but up until now, they have lacked even a meaningful definition. One consequence is that incorrect and inefficient protocols have proliferated. This paper provides the first treatment of these p ..."
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Cited by 463 (13 self)
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Entity authentication and key distribution are central cryptographic problems in distributed computing  but up until now, they have lacked even a meaningful definition. One consequence is that incorrect and inefficient protocols have proliferated. This paper provides the first treatment of these problems in the complexitytheoretic framework of modern cryptography. Addressed in detail are two problems of the symmetric, twoparty setting: mutual authentication and authenticated key exchange. For each we present a definition, protocol, and proof that the protocol meets its goal, assuming the (minimal) assumption of pseudorandom function. When this assumption is appropriately instantiated, the protocols given are practical and efficient.
Designing Programs That Check Their Work
, 1989
"... A program correctness checker is an algorithm for checking the output of a computation. That is, given a program and an instance on which the program is run, the checker certifies whether the output of the program on that instance is correct. This paper defines the concept of a program checker. It d ..."
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Cited by 307 (17 self)
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A program correctness checker is an algorithm for checking the output of a computation. That is, given a program and an instance on which the program is run, the checker certifies whether the output of the program on that instance is correct. This paper defines the concept of a program checker. It designs program checkers for a few specific and carefully chosen problems in the class FP of functions computable in polynomial time. Problems in FP for which checkers are presented in this paper include Sorting, Matrix Rank and GCD. It also applies methods of modern cryptography, especially the idea of a probabilistic interactive proof, to the design of program checkers for group theoretic computations. Two strucural theorems are proven here. One is a characterization of problems that can be checked. The other theorem establishes equivalence classes of problems such that whenever one problem in a class is checkable, all problems in the class are checkable.
Publickey Cryptosystems Provably Secure against Chosen Ciphertext Attacks
 In Proc. of the 22nd STOC
, 1995
"... We show how to construct a publickey cryptosystem (as originally defined by Diffie and Hellman) secure against chosen ciphertext attacks, given a publickey cryptosystem secure against passive eavesdropping and a noninteractive zeroknowledge proof system in the shared string model. No such secure ..."
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Cited by 249 (15 self)
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We show how to construct a publickey cryptosystem (as originally defined by Diffie and Hellman) secure against chosen ciphertext attacks, given a publickey cryptosystem secure against passive eavesdropping and a noninteractive zeroknowledge proof system in the shared string model. No such secure cryptosystems were known before. Key words. cryptography, randomized algorithms AMS subject classifications. 68M10, 68Q20, 68Q22, 68R05, 68R10 A preliminary version of this paper appeared in the Proc. of the Twenty Second ACM Symposium of Theory of Computing. y Incumbent of the Morris and Rose Goldman Career Development Chair, Dept. of Applied Mathematics and Computer Science, Weizmann Institute of Science, Rehovot 76100, Israel. Work performed while at the IBM Almaden Research Center. Research supported by an Alon Fellowship and a grant from the Israel Science Foundation administered by the Israeli Academy of Sciences. Email: naor@wisdom.weizmann.ac.il. z IBM Research Division, T.J ...
Optimal Asymmetric Encryption – How to Encrypt with RSA
, 1995
"... Given an arbitrary kbit to kbit trapdoor permutation f and a hash function, we exhibit an encryption scheme for which (i) any string x of length slightly less than k bits can be encrypted as f(rx), where rx is a simple probabilistic encoding of x depending on the hash function; and (ii) the scheme ..."
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Cited by 204 (18 self)
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Given an arbitrary kbit to kbit trapdoor permutation f and a hash function, we exhibit an encryption scheme for which (i) any string x of length slightly less than k bits can be encrypted as f(rx), where rx is a simple probabilistic encoding of x depending on the hash function; and (ii) the scheme can be proven semantically secure assuming the hash function is \ideal. " Moreover, a slightly enhanced scheme is shown to have the property that the adversary can create ciphertexts only of strings for which she \knows " the corresponding plaintextssuch ascheme is not only semantically secure but also nonmalleable and secure against chosenciphertext attack.
On the Composition of ZeroKnowledge Proof Systems
 SIAM Journal on Computing
, 1990
"... : The wide applicability of zeroknowledge interactive proofs comes from the possibility of using these proofs as subroutines in cryptographic protocols. A basic question concerning this use is whether the (sequential and/or parallel) composition of zeroknowledge protocols is zeroknowledge too. We ..."
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Cited by 190 (14 self)
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: The wide applicability of zeroknowledge interactive proofs comes from the possibility of using these proofs as subroutines in cryptographic protocols. A basic question concerning this use is whether the (sequential and/or parallel) composition of zeroknowledge protocols is zeroknowledge too. We demonstrate the limitations of the composition of zeroknowledge protocols by proving that the original definition of zeroknowledge is not closed under sequential composition; and that even the strong formulations of zeroknowledge (e.g. blackbox simulation) are not closed under parallel execution. We present lower bounds on the round complexity of zeroknowledge proofs, with significant implications to the parallelization of zeroknowledge protocols. We prove that 3round interactive proofs and constantround ArthurMerlin proofs that are blackbox simulation zeroknowledge exist only for languages in BPP. In particular, it follows that the "parallel versions" of the first interactive proo...
On Defining Proofs of Knowledge
, 1998
"... The notion of a "proof of knowledge," suggested by Gold wasset, Micali and Rackoff, has been used in many works as a tool for the construction of cryptographic protocols and other schemes. Yet the commonly cited formalizations of this notion are unsatisfactory and in particular inadequate for s ..."
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Cited by 139 (23 self)
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The notion of a "proof of knowledge," suggested by Gold wasset, Micali and Rackoff, has been used in many works as a tool for the construction of cryptographic protocols and other schemes. Yet the commonly cited formalizations of this notion are unsatisfactory and in particular inadequate for some of the applications in which they are used. Consequently,
On Hiding Information from an Oracle
, 1989
"... : We consider the problem of computing with encrypted data. Player A wishes to know the value f(x) for some x but lacks the power to compute it. Player B has the power to compute f and is willing to send f(y) to A if she sends him y, for any y. Informally, an encryption scheme for the problem f is a ..."
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Cited by 129 (15 self)
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: We consider the problem of computing with encrypted data. Player A wishes to know the value f(x) for some x but lacks the power to compute it. Player B has the power to compute f and is willing to send f(y) to A if she sends him y, for any y. Informally, an encryption scheme for the problem f is a method by which A, using her inferior resources, can transform the cleartext instance x into an encrypted instance y, obtain f(y) from B, and infer f(x) from f(y) in such a way that B cannot infer x from y. When such an encryption scheme exists, we say that f is encryptable. The framework defined in this paper enables us to prove precise statements about what an encrypted instance hides and what it leaks, in an informationtheoretic sense. Our definitions are cast in the language of probability theory and do not involve assumptions such as the intractability of factoring or the existence of oneway functions. We use our framework to describe encryption schemes for some wellknown function...
Definitions And Properties Of ZeroKnowledge Proof Systems
 Journal of Cryptology
, 1994
"... In this paper we investigate some properties of zeroknowledge proofs, a notion introduced by Goldwasser, Micali and Rackoff. We introduce and classify two definitions of zeroknowledge: auxiliary \Gamma input zeroknowledge and blackbox \Gamma simulation zeroknowledge. We explain why auxiliaryinp ..."
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Cited by 112 (10 self)
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In this paper we investigate some properties of zeroknowledge proofs, a notion introduced by Goldwasser, Micali and Rackoff. We introduce and classify two definitions of zeroknowledge: auxiliary \Gamma input zeroknowledge and blackbox \Gamma simulation zeroknowledge. We explain why auxiliaryinput zeroknowledge is a definition more suitable for cryptographic applications than the original [GMR1] definition. In particular, we show that any protocol solely composed of subprotocols which are auxiliaryinput zeroknowledge is itself auxiliaryinput zeroknowledge. We show that blackboxsimulation zeroknowledge implies auxiliaryinput zeroknowledge (which in turn implies the [GMR1] definition). We argue that all known zeroknowledge proofs are in fact blackboxsimulation zeroknowledge (i.e., were proved zeroknowledge using blackboxsimulation of the verifier). As a result, all known zeroknowledge proof systems are shown to be auxiliaryinput zeroknowledge and can be used for cryptographic applications such as those in [GMW2]. We demonstrate the triviality of certain classes of zeroknowledge proof systems, in the sense that only languages in BPP have zeroknowledge proofs of these classes. In particular, we show that any language having a Las Vegas zeroknowledge proof system necessarily belongs to RP . We show that randomness of both the verifier and the prover, and nontriviality of the interaction are essential properties of (nontrivial) auxiliaryinput zeroknowledge proofs.
On the Concurrent Composition of ZeroKnowledge Proofs
 In EuroCrypt99, Springer LNCS 1592
, 1999
"... Abstract. We examine the concurrent composition of zeroknowledge proofs. By concurrent composition, we indicate a single prover that is involved in multiple, simultaneous zeroknowledge proofs with one or multiple verifiers. Under this type of composition it is believed that standard zeroknowledge ..."
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Cited by 110 (3 self)
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Abstract. We examine the concurrent composition of zeroknowledge proofs. By concurrent composition, we indicate a single prover that is involved in multiple, simultaneous zeroknowledge proofs with one or multiple verifiers. Under this type of composition it is believed that standard zeroknowledge protocols are no longer zeroknowledge. We show that, modulo certain complexity assumptions, any statement in NP has k ɛround proofs and arguments in which one can efficiently simulate any k O(1) concurrent executions of the protocol.