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117
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 189 (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.
Undirected STConnectivity in LogSpace
, 2004
"... We present a deterministic, logspace algorithm that solves stconnectivity in undirected graphs. The previous bound on the space complexity of undirected stconnectivity was log 4/3 (·) obtained by Armoni, TaShma, Wigderson and Zhou [ATSWZ00]. As undirected stconnectivity is complete for the clas ..."
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Cited by 167 (3 self)
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We present a deterministic, logspace algorithm that solves stconnectivity in undirected graphs. The previous bound on the space complexity of undirected stconnectivity was log 4/3 (·) obtained by Armoni, TaShma, Wigderson and Zhou [ATSWZ00]. As undirected stconnectivity is complete for the class of problems solvable by symmetric, nondeterministic, logspace computations (the class SL), this algorithm implies that SL = L (where L is the class of problems solvable by deterministic logspace computations). Our algorithm also implies logspace constructible universaltraversal sequences for graphs with restricted labelling and logspace constructible universalexploration sequences for general graphs.
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 139 (21 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...
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 119 (28 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.
Extractors and Pseudorandom Generators
 Journal of the ACM
, 1999
"... We introduce a new approach to constructing extractors. Extractors are algorithms that transform a "weakly random" distribution into an almost uniform distribution. Explicit constructions of extractors have a variety of important applications, and tend to be very difficult to obtain. ..."
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Cited by 116 (6 self)
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We introduce a new approach to constructing extractors. Extractors are algorithms that transform a "weakly random" distribution into an almost uniform distribution. Explicit constructions of extractors have a variety of important applications, and tend to be very difficult to obtain.
Infeasibility of instance compression and succinct PCPs for NP
 Electronic Colloquium on Computational Complexity (ECCC
"... The ORSAT problem asks, given Boolean formulae φ1,..., φm each of size at most n, whether at least one of the φi’s is satisfiable. We show that there is no reduction from ORSAT to any set A where the length of the output is bounded by a polynomial in n, unless NP ⊆ coNP/poly, and the PolynomialTi ..."
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Cited by 67 (1 self)
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The ORSAT problem asks, given Boolean formulae φ1,..., φm each of size at most n, whether at least one of the φi’s is satisfiable. We show that there is no reduction from ORSAT to any set A where the length of the output is bounded by a polynomial in n, unless NP ⊆ coNP/poly, and the PolynomialTime Hierarchy collapses. This result settles an open problem proposed by Bodlaender et. al. [4] and Harnik and Naor [15] and has a number of implications. • A number of parametric NP problems, including Satisfiability, Clique, Dominating Set and Integer Programming, are not instance compressible or polynomially kernelizable unless NP ⊆ coNP/poly. • Satisfiability does not have PCPs of size polynomial in the number of variables unless NP ⊆ coNP/poly. • An approach of Harnik and Naor to constructing collisionresistant hash functions from oneway functions is unlikely to be viable in its present form. • (BuhrmanHitchcock) There are no subexponentialsize hard sets for NP unless NP is in coNP/poly. We also study probabilistic variants of compression, and show various results about and connections between these variants. To this end, we introduce a new strong derandomization hypothesis, the Oracle Derandomization Hypothesis, and discuss how it relates to traditional derandomization assumptions. Categories and Subject Descriptors
Extracting randomness from samplable distributions
 In Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science
, 2000
"... The standard notion of a randomness extractor is a procedure which converts any weak source of randomness into an almost uniform distribution. The conversion necessarily uses a small amount of pure randomness, which can be eliminated by complete enumeration in some, but not all, applications. Here, ..."
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Cited by 66 (7 self)
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The standard notion of a randomness extractor is a procedure which converts any weak source of randomness into an almost uniform distribution. The conversion necessarily uses a small amount of pure randomness, which can be eliminated by complete enumeration in some, but not all, applications. Here, we consider the problem of deterministically converting a weak source of randomness into an almost uniform distribution. Previously, deterministic extraction procedures were known only for sources satisfying strong independence requirements. In this paper, we look at sources which are samplable, i.e. can be generated by an efficient sampling algorithm. We seek an efficient deterministic procedure that, given a sample from any samplable distribution of sufficiently large minentropy, gives an almost uniformly distributed output. We explore the conditions under which such deterministic extractors exist. We observe that no deterministic extractor exists if the sampler is allowed to use more computational resources than the extractor. On the other hand, if the extractor is allowed (polynomially) more resources than the sampler, we show that deterministic extraction becomes possible. This is true unconditionally in the nonuniform setting (i.e., when the extractor can be computed by a small circuit), and (necessarily) relies on complexity assumptions in the uniform setting. One of our uniform constructions is as follows: assuming that there are problems in���ÌÁÅ�ÇÒthat are not solvable by subexponentialsize circuits with¦� gates, there is an efficient extractor that transforms any samplable distribution of lengthÒand minentropy Ò into an output distribution of length ÇÒ, whereis any sufficiently small constant. The running time of the extractor is polynomial inÒand the circuit complexity of the sampler. These extractors are based on a connection be
In Search of an Easy Witness: Exponential Time vs. Probabilistic Polynomial Time
, 2002
"... Restricting the search space {0, 1} n to the set of truth tables of “easy ” Boolean functions on log n variables, as well as using some known hardnessrandomness tradeoffs, we establish a number of results relating the complexity of exponentialtime and probabilistic polynomialtime complexity class ..."
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Cited by 54 (7 self)
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Restricting the search space {0, 1} n to the set of truth tables of “easy ” Boolean functions on log n variables, as well as using some known hardnessrandomness tradeoffs, we establish a number of results relating the complexity of exponentialtime and probabilistic polynomialtime complexity classes. In particular, we show that NEXP ⊂ P/poly ⇔ NEXP = MA; this can be interpreted as saying that no derandomization of MA (and, hence, of promiseBPP) is possible unless NEXP contains a hard Boolean function. We also prove several downward closure results for ZPP, RP, BPP, and MA; e.g., we show EXP = BPP ⇔ EE = BPE, where EE is the doubleexponential time class and BPE is the exponentialtime analogue of BPP.
Algebrization: A new barrier in complexity theory
 MIT Theory of Computing Colloquium
, 2007
"... Any proof of P � = NP will have to overcome two barriers: relativization and natural proofs. Yet over the last decade, we have seen circuit lower bounds (for example, that PP does not have linearsize circuits) that overcome both barriers simultaneously. So the question arises of whether there is a ..."
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Cited by 51 (3 self)
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Any proof of P � = NP will have to overcome two barriers: relativization and natural proofs. Yet over the last decade, we have seen circuit lower bounds (for example, that PP does not have linearsize circuits) that overcome both barriers simultaneously. So the question arises of whether there is a third barrier to progress on the central questions in complexity theory. In this paper we present such a barrier, which we call algebraic relativization or algebrization. The idea is that, when we relativize some complexity class inclusion, we should give the simulating machine access not only to an oracle A, but also to a lowdegree extension of A over a finite field or ring. We systematically go through basic results and open problems in complexity theory to delineate the power of the new algebrization barrier. First, we show that all known nonrelativizing results based on arithmetization—both inclusions such as IP = PSPACE and MIP = NEXP, and separations such as MAEXP � ⊂ P/poly —do indeed algebrize. Second, we show that almost all of the major open problems—including P versus NP, P versus RP, and NEXP versus P/poly—will require nonalgebrizing techniques. In some cases algebrization seems to explain exactly why progress stopped where it did: for example, why we have superlinear circuit lower bounds for PromiseMA but not for NP. Our second set of results follows from lower bounds in a new model of algebraic query complexity, which we introduce in this paper and which is interesting in its own right. Some of our lower bounds use direct combinatorial and algebraic arguments, while others stem from a surprising connection between our model and communication complexity. Using this connection, we are also able to give an MAprotocol for the Inner Product function with O ( √ n log n) communication (essentially matching a lower bound of Klauck), as well as a communication complexity conjecture whose truth would imply NL � = NP. 1
Nonuniform ACC circuit lower bounds
, 2010
"... The class ACC consists of circuit families with constant depth over unbounded fanin AND, OR, NOT, and MODm gates, where m> 1 is an arbitrary constant. We prove: • NTIME[2 n] does not have nonuniform ACC circuits of polynomial size. The size lower bound can be slightly strengthened to quasipoly ..."
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Cited by 50 (8 self)
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The class ACC consists of circuit families with constant depth over unbounded fanin AND, OR, NOT, and MODm gates, where m> 1 is an arbitrary constant. We prove: • NTIME[2 n] does not have nonuniform ACC circuits of polynomial size. The size lower bound can be slightly strengthened to quasipolynomials and other less natural functions. • ENP, the class of languages recognized in 2O(n) time with an NP oracle, doesn’t have nonuniform ACC circuits of 2no(1) size. The lower bound gives an exponential sizedepth tradeoff: for every d there is a δ> 0 such that ENP doesn’t have depthd ACC circuits of size 2nδ. Previously, it was not known whether EXP NP had depth3 polynomial size circuits made out of only MOD6 gates. The highlevel strategy is to design faster algorithms for the circuit satisfiability problem over ACC circuits, then prove that such algorithms entail the above lower bounds. The algorithm combines known properties of ACC with fast rectangular matrix multiplication and dynamic programming, while the second step requires a subtle strengthening of the author’s prior work [STOC’10]. Supported by the Josef Raviv Memorial Fellowship.