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Cryptographic Limitations on Learning Boolean Formulae and Finite Automata
 PROCEEDINGS OF THE TWENTYFIRST ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING
, 1989
"... In this paper we prove the intractability of learning several classes of Boolean functions in the distributionfree model (also called the Probably Approximately Correct or PAC model) of learning from examples. These results are representation independent, in that they hold regardless of the syntact ..."
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Cited by 347 (15 self)
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In this paper we prove the intractability of learning several classes of Boolean functions in the distributionfree model (also called the Probably Approximately Correct or PAC model) of learning from examples. These results are representation independent, in that they hold regardless of the syntactic form in which the learner chooses to represent its hypotheses. Our methods reduce the problems of cracking a number of wellknown publickey cryptosystems to the learning problems. We prove that a polynomialtime learning algorithm for Boolean formulae, deterministic finite automata or constantdepth threshold circuits would have dramatic consequences for cryptography and number theory: in particular, such an algorithm could be used to break the RSA cryptosystem, factor Blum integers (composite numbers equivalent to 3 modulo 4), and detect quadratic residues. The results hold even if the learning algorithm is only required to obtain a slight advantage in prediction over random guessing. The techniques used demonstrate an interesting duality between learning and cryptography. We also apply our results to obtain strong intractability results for approximating a generalization of graph coloring.
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 284 (20 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 ...
Bit Commitment Using PseudoRandomness
 Journal of Cryptology
, 1991
"... We show how a pseudorandom generator can provide a bit commitment protocol. We also analyze the number of bits communicated when parties commit to many bits simultaneously, and show that the assumption of the existence of pseudorandom generators suffices to assure amortized O(1) bits of communicat ..."
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Cited by 282 (16 self)
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We show how a pseudorandom generator can provide a bit commitment protocol. We also analyze the number of bits communicated when parties commit to many bits simultaneously, and show that the assumption of the existence of pseudorandom generators suffices to assure amortized O(1) bits of communication per bit commitment.
How to Timestamp a Digital Document
 Journal of Cryptology
, 1991
"... The prospect of a world in which all text, audio, picture, and video documents are in digital form on easily modifiable media raises the issue of how to certify when a document was created or last changed. The problem is to timestamp the data, not the medium. We propose computationally practical ..."
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Cited by 247 (5 self)
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The prospect of a world in which all text, audio, picture, and video documents are in digital form on easily modifiable media raises the issue of how to certify when a document was created or last changed. The problem is to timestamp the data, not the medium. We propose computationally practical procedures for digital timestamping of such documents so that it is infeasible for a user either to backdate or to forwarddate his document, even with the collusion of a timestamping service. Our procedures maintain complete privacy of the documents themselves, and require no recordkeeping by the timestamping service. Appeared, with minor editorial changes, in Journal of Cryptology, Vol. 3, No. 2, pp. 99111, 1991. 0 Time's glory is to calm contending kings, To unmask falsehood, and bring truth to light, To stamp the seal of time in aged things, To wake the morn, and sentinel the night, To wrong the wronger till he render right. The Rape of Lucrece, l. 941 1 Introduction ...
Pseudorandom generators for spacebounded computation
 Combinatorica
, 1992
"... Pseudorandom generators are constructed which convert O(SlogR) truly random bits to R bits that appear random to any algorithm that runs in SPACE(S). In particular, any randomized polynomial time algorithm that runs in space S can be simulated using only O(Slogn) random bits. An application of these ..."
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Cited by 245 (11 self)
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Pseudorandom generators are constructed which convert O(SlogR) truly random bits to R bits that appear random to any algorithm that runs in SPACE(S). In particular, any randomized polynomial time algorithm that runs in space S can be simulated using only O(Slogn) random bits. An application of these generators is an explicit construction of universal traversal sequences (for arbitrary graphs) of length n O(l~ The generators constructed are technically stronger than just appearing random to spacebounded machines, and have several other applications. In particular, applications are given for "deterministic amplification " (i.e. reducing the probability of error of randomized algorithms), as well as generalizations of it. 1.
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 228 (20 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. &quot; Moreover, a slightly enhanced scheme is shown to have the property that the adversary can create ciphertexts only of strings for which she \knows &quot; the corresponding plaintextssuch ascheme is not only semantically secure but also nonmalleable and secure against chosenciphertext attack.
Limits on the Provable Consequences of Oneway Permutations
, 1989
"... We present strong evidence that the implication, "if oneway permutations exist, then secure secret key agreement is possible" is not provable by standard techniques. Since both sides of this implication are widely believed true in real life, to show that the implication is false requir ..."
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Cited by 205 (0 self)
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We present strong evidence that the implication, "if oneway permutations exist, then secure secret key agreement is possible" is not provable by standard techniques. Since both sides of this implication are widely believed true in real life, to show that the implication is false requires a new model. We consider a world where dl parties have access to a black box or a randomly selected permutation. Being totally random, this permutation will be strongly oneway in provable, informationthevretic way. We show that, if P = NP, no protocol for secret key agreement is secure in such setting. Thus, to prove that a secret key greement protocol which uses a oneway permutation as a black box is secure is as hrd as proving F NP. We also obtain, as corollary, that there is an oracle relative to which the implication is false, i.e., there is a oneway permutation, yet secretexchange is impossible. Thus, no technique which relativizes can prove that secret exchange can be based on any oneway permutation. Our results present a general framework for proving statements of the form, "Cryptographic application X is not likely possible based solely on complexity assumption Y." 1
Derandomizing Polynomial Identity Tests Means Proving Circuit Lower Bounds (Extended Abstract)
, 2003
"... Since Polynomial Identity Testing is a coRP problem, we obtain the following corollary: If RP = P (or, even, coRP ` "ffl?0NTIME(2nffl), infinitely often), then NEXP is not computable by polynomialsize arithmetic circuits. Thus, establishing that RP = coRP or BPP = P would require proving s ..."
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Cited by 187 (4 self)
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Since Polynomial Identity Testing is a coRP problem, we obtain the following corollary: If RP = P (or, even, coRP ` &quot;ffl?0NTIME(2nffl), infinitely often), then NEXP is not computable by polynomialsize arithmetic circuits. Thus, establishing that RP = coRP or BPP = P would require proving superpolynomial lower bounds for Boolean or arithmetic circuits. We also show that any derandomization of RNC would yield new circuit lower bounds for a language in NEXP.
Numbertheoretic constructions of efficient pseudorandom functions
 In 38th Annual Symposium on Foundations of Computer Science
, 1997
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