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109
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 173 (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.
Unbalanced expanders and randomness extractors from parvareshvardy codes
 In Proceedings of the 22nd Annual IEEE Conference on Computational Complexity
, 2007
"... We give an improved explicit construction of highly unbalanced bipartite expander graphs with expansion arbitrarily close to the degree (which is polylogarithmic in the number of vertices). Both the degree and the number of righthand vertices are polynomially close to optimal, whereas the previous ..."
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Cited by 120 (7 self)
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We give an improved explicit construction of highly unbalanced bipartite expander graphs with expansion arbitrarily close to the degree (which is polylogarithmic in the number of vertices). Both the degree and the number of righthand vertices are polynomially close to optimal, whereas the previous constructions of TaShma, Umans, and Zuckerman (STOC ‘01) required at least one of these to be quasipolynomial in the optimal. Our expanders have a short and selfcontained description and analysis, based on the ideas underlying the recent listdecodable errorcorrecting codes of Parvaresh and Vardy (FOCS ‘05). Our expanders can be interpreted as nearoptimal “randomness condensers, ” that reduce the task of extracting randomness from sources of arbitrary minentropy rate to extracting randomness from sources of minentropy rate arbitrarily close to 1, which is a much easier task. Using this connection, we obtain a new construction of randomness extractors that is optimal up to constant factors, while being much simpler than the previous construction of Lu et al. (STOC ‘03) and improving upon it when the error parameter is small (e.g. 1/poly(n)).
Lossless condensers, unbalanced expanders, and extractors
 In Proceedings of the 33rd Annual ACM Symposium on Theory of Computing
, 2001
"... Abstract Trevisan showed that many pseudorandom generator constructions give rise to constructionsof explicit extractors. We show how to use such constructions to obtain explicit lossless condensers. A lossless condenser is a probabilistic map using only O(log n) additional random bitsthat maps n bi ..."
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Cited by 98 (17 self)
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Abstract Trevisan showed that many pseudorandom generator constructions give rise to constructionsof explicit extractors. We show how to use such constructions to obtain explicit lossless condensers. A lossless condenser is a probabilistic map using only O(log n) additional random bitsthat maps n bits strings to poly(log K) bit strings, such that any source with support size Kis mapped almost injectively to the smaller domain. Our construction remains the best lossless condenser to date.By composing our condenser with previous extractors, we obtain new, improved extractors. For small enough minentropies our extractors can output all of the randomness with only O(log n) bits. We also obtain a new disperser that works for every entropy loss, uses an O(log n)bit seed, and has only O(log n) entropy loss. This is the best disperser construction to date,and yields other applications. Finally, our lossless condenser can be viewed as an unbalanced
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 65 (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
Pseudorandom generators for all hardness
 Journal of Computer and System Science
, 2003
"... A pseudorandom generator (PRG) is a function that “stretches ” a short random seed into a longer pseudorandom output string that “fools ” small circuits: Definition 1 (ɛPRG) An ɛPRG for size s is a function G: {0, 1} t →{0, 1} m such that for all circuits C of size at most s:  Pr[C(G(z))=1]−Pr[ ..."
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Cited by 58 (8 self)
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A pseudorandom generator (PRG) is a function that “stretches ” a short random seed into a longer pseudorandom output string that “fools ” small circuits: Definition 1 (ɛPRG) An ɛPRG for size s is a function G: {0, 1} t →{0, 1} m such that for all circuits C of size at most s:  Pr[C(G(z))=1]−Pr[C(x) =1]≤ɛ. z x PRGs entail hard functions, so (in the absence of strong circuit lower bounds) they are constructed using the assumption that hard functions exist. They can therefore be seen as objects that convert computational hardness into pseudorandomness. We construct the first pseudorandom generators with logarithmic seed length that convert s bits of hardness
Extractors: Optimal up to Constant Factors
 STOC'03
, 2003
"... This paper provides the first explicit construction of extractors which are simultaneously optimal up to constant factors in both seed length and output length. More precisely, for every n, k, our extractor uses a random seed of length O(log n) to transform any random source on n bits with (min)ent ..."
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Cited by 50 (12 self)
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This paper provides the first explicit construction of extractors which are simultaneously optimal up to constant factors in both seed length and output length. More precisely, for every n, k, our extractor uses a random seed of length O(log n) to transform any random source on n bits with (min)entropy k, into a distribution on (1 − α)k bits that is ɛclose to uniform. Here α and ɛ can be taken to be any positive constants. (In fact, ɛ can be almost polynomially small). Our improvements are obtained via three new techniques, each of which may be of independent interest. The first is a general construction of mergers [22] from locally decodable errorcorrecting codes. The second introduces new condensers that have constant seed length (and retain a constant fraction of the minentropy in the random source). The third is a way to augment the “winwin repeated condensing” paradigm of [17] with error reduction techniques like [15] so that the our constant seedlength condensers can be used without error accumulation.
Pseudorandomness and averagecase complexity via uniform reductions
 IN PROCEEDINGS OF THE 17TH ANNUAL IEEE CONFERENCE ON COMPUTATIONAL COMPLEXITY
, 2002
"... Impagliazzo and Wigderson (36th FOCS, 1998) gave the first construction of pseudorandom generators from a uniform complexity assumption on EXP (namely EXP � = BPP). Unlike results in the nonuniform setting, their result does not provide a continuous tradeoff between worstcase hardness and pseudor ..."
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Cited by 50 (7 self)
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Impagliazzo and Wigderson (36th FOCS, 1998) gave the first construction of pseudorandom generators from a uniform complexity assumption on EXP (namely EXP � = BPP). Unlike results in the nonuniform setting, their result does not provide a continuous tradeoff between worstcase hardness and pseudorandomness, nor does it explicitly establish an averagecase hardness result. In this paper: ◦ We obtain an optimal worstcase to averagecase connection for EXP: if EXP � ⊆ BPTIME(t(n)), then EXP has problems that cannot be solved on a fraction 1/2 + 1/t ′ (n) of the inputs by BPTIME(t ′ (n)) algorithms, for t ′ = t Ω(1). ◦ We exhibit a PSPACEcomplete selfcorrectible and downward selfreducible problem. This slightly simplifies and strengthens the proof of Impagliazzo and Wigderson, which used a #Pcomplete problem with these properties. ◦ We argue that the results of Impagliazzo and Wigderson, and the ones in this paper, cannot be proved via “blackbox” uniform reductions.
Extractor Codes
, 2001
"... We de ne new error correcting codes based on extractors. Weshow that for certain choices of parameters these codes have better list decoding properties than are known for other codes, and are provably better than ReedSolomon codes. We further show that codes with strong list decoding properties ar ..."
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Cited by 47 (6 self)
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We de ne new error correcting codes based on extractors. Weshow that for certain choices of parameters these codes have better list decoding properties than are known for other codes, and are provably better than ReedSolomon codes. We further show that codes with strong list decoding properties are equivalent to slice extractors, a variant of extractors. Wegive an application of extractor codes to extracting many hardcore bits from a oneway function, using few auxiliary random bits. Finally,weshow that explicit slice extractors for certain other parameters would yield optimal bipartite Ramsey graphs.
Deterministic Extractors for Affine Sources over Large Fields
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 108 (2005)
, 2005
"... An (n, k)affine source over a finite field F is a random variable X = (X1,..., Xn) ∈ Fn, which is uniformly distributed over an (unknown) kdimensional affine subspace of F n. We show how to (deterministically) extract practically all the randomness from affine sources, for any field of size large ..."
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Cited by 44 (6 self)
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An (n, k)affine source over a finite field F is a random variable X = (X1,..., Xn) ∈ Fn, which is uniformly distributed over an (unknown) kdimensional affine subspace of F n. We show how to (deterministically) extract practically all the randomness from affine sources, for any field of size larger than n c (where c is a large enough constant). Our main results are as follows: 1. (For arbitrary k): For any n, k and any F of size larger than n 20, we give an explicit construction for a function D: F n → F k−1, such that for any (n, k)affine source X over F, the distribution of D(X) is ɛclose to uniform, where ɛ is polynomially small in F. 2. (For k = 1): For any n and any F of size larger than n c, we give an explicit construction for a function D: F n → {0, 1} (1−δ) log 2 F  , such that for any (n, 1)affine source X over F, the distribution of D(X) is ɛclose to uniform, where ɛ is polynomially small in F. Here, δ> 0 is an arbitrary small constant, and c is a constant depending on δ.