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Simple Extractors for All MinEntropies and a New PseudoRandom Generator
 Journal of the ACM
, 2001
"... A “randomness extractor ” is an algorithm that given a sample from a distribution with sufficiently high minentropy and a short random seed produces an output that is statistically indistinguishable from uniform. (Minentropy is a measure of the amount of randomness in a distribution). We present a ..."
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Cited by 112 (30 self)
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A “randomness extractor ” is an algorithm that given a sample from a distribution with sufficiently high minentropy and a short random seed produces an output that is statistically indistinguishable from uniform. (Minentropy is a measure of the amount of randomness in a distribution). We present a simple, selfcontained extractor construction that produces good extractors for all minentropies. Our construction is algebraic and builds on a new polynomialbased approach introduced by TaShma, Zuckerman, and Safra [TSZS01]. Using our improvements, we obtain, for example, an extractor with output length m = k/(log n) O(1/α) and seed length (1 + α) log n for an arbitrary 0 < α ≤ 1, where n is the input length, and k is the minentropy of the input distribution. A “pseudorandom generator ” is an algorithm that given a short random seed produces a long output that is computationally indistinguishable from uniform. Our technique also gives a new way to construct pseudorandom generators from functions that require large circuits. Our pseudorandom generator construction is not based on the NisanWigderson generator [NW94], and turns worstcase hardness directly into pseudorandomness. The parameters of our generator match those in [IW97, STV01] and in particular are strong enough to obtain a new proof that P = BP P if E requires exponential size circuits.
Derandomizing ArthurMerlin Games using Hitting Sets
, 1999
"... We prove that AM (and hence Graph Nonisomorphism) is in NP if for some > 0, some language in NE \ coNE requires nondeterministic circuits of size 2 n . This improves recent results of Arvind and K obler and of Klivans and Van Melkebeek who proved the same conclusion, but under stronger hardnes ..."
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Cited by 67 (0 self)
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We prove that AM (and hence Graph Nonisomorphism) is in NP if for some > 0, some language in NE \ coNE requires nondeterministic circuits of size 2 n . This improves recent results of Arvind and K obler and of Klivans and Van Melkebeek who proved the same conclusion, but under stronger hardness assumptions, namely, either the existence of a language in NE \ coNE which cannot be approximated by nondeterministic circuits of size less than 2 n or the existence of a language in NE \ coNE which requires oracle circuits of size 2 n with oracle gates for SAT (satisfiability). The previous results on derandomizing AM were based on pseudorandom generators. In contrast, our approach is based on a strengthening of Andreev, Clementi and Rolim's hitting set approach to derandomization. As a spinoff, we show that this approach is strong enough to give an easy (if the existence of explicit dispersers can be assumed known) proof of the following implication: For some > 0, if there is a l...
Easiness Assumptions and Hardness Tests: Trading Time for Zero Error
 Journal of Computer and System Sciences
, 2000
"... We propose a new approach towards derandomization in the uniform setting, where it is computationally hard to nd possible mistakes in the simulation of a given probabilistic algorithm. The approach consists in combining both easiness and hardness complexity assumptions: if a derandomization metho ..."
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Cited by 43 (2 self)
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We propose a new approach towards derandomization in the uniform setting, where it is computationally hard to nd possible mistakes in the simulation of a given probabilistic algorithm. The approach consists in combining both easiness and hardness complexity assumptions: if a derandomization method based on an easiness assumption fails, then we obtain a certain hardness test that can be used to remove error in BPP algorithms. As an application, we prove that every RP algorithm can be simulated by a zeroerror probabilistic algorithm, running in expected subexponential time, that appears correct innitely often (i.o.) to every ecient adversary. A similar result by Impagliazzo and Wigderson (FOCS'98) states that BPP allows deterministic subexponentialtime simulations that appear correct with respect to any eciently sampleable distribution i.o., under the assumption that EXP 6= BPP; in contrast, our result does not rely on any unproven assumptions. As another application of our...
Circuit Minimization Problem
 In ACM Symposium on Theory of Computing (STOC
, 1999
"... We study the complexity of the circuit minimization problem: given the truth table of a Boolean function f and a parameter s, decide whether f can be realized by a Boolean circuit of size at most s. We argue why this problem is unlikely to be in P (or even in P=poly) by giving a number of surpris ..."
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Cited by 32 (1 self)
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We study the complexity of the circuit minimization problem: given the truth table of a Boolean function f and a parameter s, decide whether f can be realized by a Boolean circuit of size at most s. We argue why this problem is unlikely to be in P (or even in P=poly) by giving a number of surprising consequences of such an assumption. We also argue that proving this problem to be NPcomplete (if it is indeed true) would imply proving strong circuit lower bounds for the class E, which appears beyond the currently known techniques. Keywords: hard Boolean functions, derandomization, natural properties, NPcompleteness. 1 Introduction An nvariable Boolean function f n : f0; 1g n ! f0; 1g can be given by either its truth table of size 2 n , or a Boolean circuit whose size may be significantly smaller than 2 n . It is well known that most Boolean functions on n variables have circuit complexity at least 2 n =n [Sha49], but so far no family of sufficiently hard functions has ...
Reconstructive dispersers and hitting set generators
 In APPROXRANDOM
, 2005
"... Abstract. We give a generic construction of an optimal hitting set generator (HSG) from any good “reconstructive ” disperser. Past constructions of optimal HSGs have been based on such disperser constructions, but have had to modify the construction in a complicated way to meet the stringent efficie ..."
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Cited by 6 (3 self)
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Abstract. We give a generic construction of an optimal hitting set generator (HSG) from any good “reconstructive ” disperser. Past constructions of optimal HSGs have been based on such disperser constructions, but have had to modify the construction in a complicated way to meet the stringent efficiency requirements of HSGs. The construction in this paper uses existing disperser constructions with the “easiest ” parameter setting in a blackbox fashion to give new constructions of optimal HSGs without any additional complications. Our results show that a straightforward composition of the NisanWigderson pseudorandom generator that is similar to the composition in works by Impagliazzo, Shaltiel and Wigderson in fact yields optimal HSGs (in contrast to the “nearoptimal ” HSGs constructed in those works). Our results also give optimal HSGs that do not use any form of hardness amplification or implicit listdecoding – like Trevisan’s extractor, the only ingredients are combinatorial designs and any good listdecodable errorcorrecting code. 1
Derandomizing ArthurMerlin Games using Hitting Sets
 Appears in Shmoys, editor, The Eleventh Annual ACMSIAM Symposium on Discrete Algorithms, SODA '00 Proceedings, 2000, pages 487493. Journal version in Journal of Algorithms, 41(1):6985, 2001, with the title Deterministic Dictionaries. RS9947 Peter B
, 1999
"... We prove that AM (and hence Graph Nonisomorphism) is in NP if for some # > 0, some language in NE coNE requires nondeterministic . This improves recent results of Arvind and Kobler and of Klivans and Van Melkebeek who proved the same conclusion, but under stronger hardness assumptions, n ..."
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Cited by 1 (0 self)
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We prove that AM (and hence Graph Nonisomorphism) is in NP if for some # > 0, some language in NE coNE requires nondeterministic . This improves recent results of Arvind and Kobler and of Klivans and Van Melkebeek who proved the same conclusion, but under stronger hardness assumptions, namely, either the existence of a language in NE coNE which cannot be approximated by nondeterministic circuits of size less than 2 or the existence of a language in NE coNE which requires oracle circuits of size 2 with oracle gates for SAT (satisfiability).
Quantum Computation Relative to Oracles (Preliminary Version)
, 2000
"... Abstract. The study of the power and limitations of quantum computation remains a major challenge in complexity theory. Key questions revolve around the quantum complexity classes EQP, BQP, NQP and their derivatives. This paper presents new relativized worlds in which (i) coRP � NQE, (ii) P = BQP a ..."
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Abstract. The study of the power and limitations of quantum computation remains a major challenge in complexity theory. Key questions revolve around the quantum complexity classes EQP, BQP, NQP and their derivatives. This paper presents new relativized worlds in which (i) coRP � NQE, (ii) P = BQP and UP = EXP, (iii) P = EQP and RP = EXP, and (iv) EQP � Σ P 2 ∪ Π P 2. We also show a partial answer to the question of whether AlmostBQP = BQP. §1. Introduction. A major question in quantum complexity theory is the power and limitations of a quantum computer for solving intractable problems. Since its inception by Benioff [6] and Feynman [14], the idea of a quantum mechanical computer was investigated in various early works by Deutsch, Jozsa, and others [13,7,8, 30,1], culminating in the seminal discovery by Shor [31] on
On Circuit Complexity Classes and Iterated Matrix Multiplication
, 2012
"... In this thesis, we study small, yet important, circuit complexity classes within NC¹, such as ACC⁰ and TC⁰00 0. We also investigate the power of a closely related problem called Iterated Matrix Multiplication and its implications in low levels of algebraic complexity theory. More concretely, • We sh ..."
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In this thesis, we study small, yet important, circuit complexity classes within NC¹, such as ACC⁰ and TC⁰00 0. We also investigate the power of a closely related problem called Iterated Matrix Multiplication and its implications in low levels of algebraic complexity theory. More concretely, • We show that extremely modestsounding lower bounds for certain problems can lead to nontrivial derandomization results. – If the word problem over S5 requires constantdepth threshold circuits of size n1+ɛ for some ɛ> 0, then any language accepted by uniform polynomialsize probabilistic threshold circuits can be solved in subexponential time (and more strongly, can be accepted by a uniform family of deterministic constantdepth threshold circuits of subexponential size.) – If there are no constantdepth arithmetic circuits of size n1+ɛ for the problem of multiplying a sequence of n 3by3 matrices, then for every constant d, blackbox identity testing for depthd arithmetic circuits with bounded individual degree can be performed in subexponential time (and even by a uniform family of deterministic constantdepth AC circuits of subexponential size).
Lecture Notes on Computational Complexity
, 2004
"... Contents 1 Introduction, P and NP 7 1.1 Computational Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 P and NP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 NPcompleteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 ..."
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Contents 1 Introduction, P and NP 7 1.1 Computational Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 P and NP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 NPcompleteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.1 Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.2 NPcompleteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.3 An NPcomplete problem . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3.4 The Problem SAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 SpaceBounded Complexity Classes 14 2.1 SpaceBounded Complexity Classes . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Reductions in NL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 NL Completeness . . . . . . . . . . . . . . . . . . . . . . . . .