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104
Free Bits, PCPs and NonApproximability  Towards Tight Results
, 1996
"... This paper continues the investigation of the connection between proof systems and approximation. The emphasis is on proving tight nonapproximability results via consideration of measures like the "free bit complexity" and the "amortized free bit complexity" of proof systems. ..."
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Cited by 203 (41 self)
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This paper continues the investigation of the connection between proof systems and approximation. The emphasis is on proving tight nonapproximability results via consideration of measures like the "free bit complexity" and the "amortized free bit complexity" of proof systems.
The NPcompleteness column: an ongoing guide
 Journal of Algorithms
, 1985
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & ..."
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Cited by 196 (0 self)
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
Unbiased Bits from Sources of Weak Randomness and Probabilistic Communication Complexity
, 1988
"... , Introduction and References only) Benny Chor Oded Goldreich MIT \Gamma Laboratory for Computer Science Cambridge, Massachusetts 02139 ABSTRACT \Gamma A new model for weak random physical sources is presented. The new model strictly generalizes previous models (e.g. the Santha and Vazirani model [2 ..."
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Cited by 184 (5 self)
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, Introduction and References only) Benny Chor Oded Goldreich MIT \Gamma Laboratory for Computer Science Cambridge, Massachusetts 02139 ABSTRACT \Gamma A new model for weak random physical sources is presented. The new model strictly generalizes previous models (e.g. the Santha and Vazirani model [24]). The sources considered output strings according to probability distributions in which no single string is too probable. The new model provides a fruitful viewpoint on problems studied previously as: ffl Extracting almost perfect bits from sources of weak randomness: the question of possibility as well as the question of efficiency of such extraction schemes are addressed. ffl Probabilistic Communication Complexity: it is shown that most functions have linear communication complexity in a very strong probabilistic sense. ffl Robustness of BPP with respect to sources of weak randomness (generalizing a result of Vazirani and Vazirani [27]). The paper has appeared in SIAM Journal o...
Pseudorandom generators without the XOR Lemma
, 1998
"... Madhu Sudan y Luca Trevisan z Salil Vadhan x Abstract Impagliazzo and Wigderson [IW97] have recently shown that if there exists a decision problem solvable in time 2 O(n) and having circuit complexity 2 n) (for all but finitely many n) then P = BPP. This result is a culmination of a serie ..."
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Cited by 128 (20 self)
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Madhu Sudan y Luca Trevisan z Salil Vadhan x Abstract Impagliazzo and Wigderson [IW97] have recently shown that if there exists a decision problem solvable in time 2 O(n) and having circuit complexity 2 n) (for all but finitely many n) then P = BPP. This result is a culmination of a series of works showing connections between the existence of hard predicates and the existence of good pseudorandom generators. The construction of Impagliazzo and Wigderson goes through three phases of "hardness amplification" (a multivariate polynomial encoding, a first derandomized XOR Lemma, and a second derandomized XOR Lemma) that are composed with the Nisan Wigderson [NW94] generator. In this paper we present two different approaches to proving the main result of Impagliazzo and Wigderson. In developing each approach, we introduce new techniques and prove new results that could be useful in future improvements and/or applications of hardnessrandomness tradeoffs. Our first result is that when (a modified version of) the NisanWigderson generator construction is applied with a "mildly" hard predicate, the result is a generator that produces a distribution indistinguishable from having large minentropy. An extractor can then be used to produce a distribution computationally indistinguishable from uniform. This is the first construction of a pseudorandom generator that works with a mildly hard predicate without doing hardness amplification. We then show that in the ImpagliazzoWigderson construction only the first hardnessamplification phase (encoding with multivariate polynomial) is necessary, since it already gives the required averagecase hardness. We prove this result by (i) establishing a connection between the hardnessamplification problem and a listdecoding...
What Can Be Computed Locally?
 SIAM J. Comput
, 1993
"... . The purpose of this paper is a study of computation that can be done locally in a distributed network, where "locally" means within time (or distance) independent of the size of the network. Locally Checkable Labeling (LCL) problems are considered, where the legality of a labeling can be ..."
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Cited by 121 (1 self)
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. The purpose of this paper is a study of computation that can be done locally in a distributed network, where "locally" means within time (or distance) independent of the size of the network. Locally Checkable Labeling (LCL) problems are considered, where the legality of a labeling can be checked locally (e.g., coloring). The results include the following: ffl There are nontrivial LCL problems that have local algorithms. ffl There is a variant of the dining philosophers problem that can be solved locally. ffl Randomization cannot make an LCL problem local; i.e., if a problem has a local randomized algorithm then it has a local deterministic algorithm. ffl It is undecidable, in general, whether a given LCL has a local algorithm. ffl However, it is decidable whether a given LCL has an algorithm that operates in a given time t. ffl Any LCL problem that has a local algorithm has one that is orderinvariant (the algorithm depends only on the order of the processor id's). Keywords: ...
BPP has Subexponential Time Simulations unless EXPTIME has Publishable Proofs (Extended Abstract)
, 1993
"... ) L'aszl'o Babai Noam Nisan y Lance Fortnow z Avi Wigderson University of Chicago Hebrew University Abstract We show that BPP can be simulated in subexponential time for infinitely many input lengths unless exponential time ffl collapses to the second level of the polynomialtime ..."
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Cited by 114 (9 self)
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) L'aszl'o Babai Noam Nisan y Lance Fortnow z Avi Wigderson University of Chicago Hebrew University Abstract We show that BPP can be simulated in subexponential time for infinitely many input lengths unless exponential time ffl collapses to the second level of the polynomialtime hierarchy, ffl has polynomialsize circuits and ffl has publishable proofs (EXPTIME=MA). We also show that BPP is contained in subexponential time unless exponential time has publishable proofs for infinitely many input lengths. In addition, we show BPP can be simulated in subexponential time for infinitely many input lengths unless there exist unary languages in MA n P . The proofs are based on the recent characterization of the power of multiprover interactive protocols and on random selfreducibility via low degree polynomials. They exhibit an interplay between Boolean circuit simulation, interactive proofs and classical complexity classes. An important feature of this proof is that it does not ...
Graph Nonisomorphism Has Subexponential Size Proofs Unless The PolynomialTime Hierarchy Collapses
 SIAM Journal on Computing
, 1998
"... We establish hardness versus randomness tradeoffs for a broad class of randomized procedures. In particular, we create efficient nondeterministic simulations of bounded round ArthurMerlin games using a language in exponential time that cannot be decided by polynomial size oracle circuits with acce ..."
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Cited by 108 (6 self)
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We establish hardness versus randomness tradeoffs for a broad class of randomized procedures. In particular, we create efficient nondeterministic simulations of bounded round ArthurMerlin games using a language in exponential time that cannot be decided by polynomial size oracle circuits with access to satisfiability. We show that every language with a bounded round ArthurMerlin game has subexponential size membership proofs for infinitely many input lengths unless exponential time coincides with the third level of the polynomialtime hierarchy (and hence the polynomialtime hierarchy collapses). This provides the first strong evidence that graph nonisomorphism has subexponential size proofs. We set up a general framework for derandomization which encompasses more than the traditional model of randomized computation. For a randomized procedure to fit within this framework, we only require that for any fixed input the complexity of checking whether the procedure succeeds on a given ...
Searchable symmetric encryption: improved definitions and efficient constructions
 Proceedings of the 13th ACM conference on Computer and communications security, CCS ’06, ACM
, 2006
"... Searchable symmetric encryption (SSE) allows a party to outsource the storage of his data to another party in a private manner, while maintaining the ability to selectively search over it. This problem has been the focus of active research and several security definitions and constructions have been ..."
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Cited by 66 (4 self)
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Searchable symmetric encryption (SSE) allows a party to outsource the storage of his data to another party in a private manner, while maintaining the ability to selectively search over it. This problem has been the focus of active research and several security definitions and constructions have been proposed. In this paper we begin by reviewing existing notions of security and propose new and stronger security definitions. We then present two constructions that we show secure under our new definitions. Interestingly, in addition to satisfying stronger security guarantees, our constructions are more efficient than all previous constructions. Further, prior work on SSE only considered the setting where only the owner of the data is capable of submitting search queries. We consider the natural extension where an arbitrary group of parties other than the owner can submit search queries. We formally define SSE in this multiuser setting, and present an efficient construction. 1
Hardness of Approximating the Minimum Distance of a Linear Code
, 2003
"... We show that the minimum distance d of a linear code is not approximable to within anyconstant factor in random polynomial time (RP), unless NP (nondeterministic polynomial time) equals RP. We also show that the minimum distance is not approximable to within an additiveerror that is linear in the b ..."
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Cited by 54 (7 self)
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We show that the minimum distance d of a linear code is not approximable to within anyconstant factor in random polynomial time (RP), unless NP (nondeterministic polynomial time) equals RP. We also show that the minimum distance is not approximable to within an additiveerror that is linear in the block length n of the code. Under the stronger assumption that NPis not contained in RQP (random quasipolynomial time), we show that the minimum distance is not approximable to within the factor 2log 1ffl(n), for any ffl> 0. Our results hold for codes over any finite field, including binary codes. In the process we show that it is hard to findapproximately nearest codewords even if the number of errors exceeds the unique decoding radius d/2 by only an arbitrarily small fraction ffld. We also prove the hardness of the nearestcodeword problem for asymptotically good codes, provided the number of errors exceeds (2
The probabilistic method yields deterministic parallel algorithms
 Proceedings of the 30th Symposium on Foundations of Computer Science, IEEE
, 1989
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