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94
Improved lowdegree testing and its applications
 IN 29TH STOC
, 1997
"... NP = PCP(log n, 1) and related results crucially depend upon the close connection betsveen the probability with which a function passes a low degree test and the distance of this function to the nearest degree d polynomial. In this paper we study a test proposed by Rubinfeld and Sudan [29]. The stro ..."
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Cited by 145 (18 self)
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NP = PCP(log n, 1) and related results crucially depend upon the close connection betsveen the probability with which a function passes a low degree test and the distance of this function to the nearest degree d polynomial. In this paper we study a test proposed by Rubinfeld and Sudan [29]. The strongest previously known connection for this test states that a function passes the test with probability 6 for some d> 7/8 iff the function has agreement N 6 with a polynomial of degree d. We presenta new, and surprisingly strong,analysiswhich shows thatthepreceding statementis truefor 6<<0.5. The analysis uses a version of Hilbe?l irreducibility, a tool used in the factoring of multivariate polynomials. As a consequence we obtain an alternate construction for the following proof system: A constant prover lround proof system for NP languages in which the verifier uses O(log n) random bits, receives answers of size O(log n) bits, and has an error probability of at most 2 – 10g*‘’. Such a proof system, which implies the NPhardness of approximating Set Cover to within fl(log n) factors, has already been obtained by Raz and Safra [28]. Our result was completed after we heard of their claim. A second consequence of our analysis is a self testerlcorrector for any buggy program that (supposedly) computes a polynomial over a finite field. If the program is correct only on 6 fraction of inputs where 15<<0.5, then the tester/corrector determines J and generates 0(~) randomized programs, such that one of the programs is correct on every input, with high probability.
Exponential lower bound for 2query locally decodable codes via a quantum argument
 Journal of Computer and System Sciences
, 2003
"... Abstract A locally decodable code encodes nbit strings x in mbit codewords C(x) in such a way that one can recover any bit xi from a corrupted codeword by querying only a few bits of that word. We use a quantum argument to prove that LDCs with 2 classical queries require exponential length: m = 2 ..."
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Cited by 123 (18 self)
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Abstract A locally decodable code encodes nbit strings x in mbit codewords C(x) in such a way that one can recover any bit xi from a corrupted codeword by querying only a few bits of that word. We use a quantum argument to prove that LDCs with 2 classical queries require exponential length: m = 2 \Omega (n). Previously this was known only for linear codes (Goldreich et al. 02). The
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 ...
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 107 (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.
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 89 (20 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
Learning polynomials with queries: The highly noisy case
, 1995
"... Given a function f mapping nvariate inputs from a finite Kearns et. al. [21] (see also [27, 28, 22]). In the setting of agfieldFintoF, we consider the task of reconstructing a list nostic learning, the learner is to make no assumptions regarding of allnvariate degreedpolynomials which agree withf ..."
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Cited by 87 (18 self)
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Given a function f mapping nvariate inputs from a finite Kearns et. al. [21] (see also [27, 28, 22]). In the setting of agfieldFintoF, we consider the task of reconstructing a list nostic learning, the learner is to make no assumptions regarding of allnvariate degreedpolynomials which agree withfon a the natural phenomena underlying the input/output relationship tiny but nonnegligible fraction, , of the input space. We give a of the function, and the goal of the learner is to come up with a randomized algorithm for solving this task which accessesfas a simple explanation which best fits the examples. Therefore the black box and runs in time polynomial in1;nand exponential in best explanation may account for only part of the phenomena. d, provided is(pd=jFj). For the special case whend=1, In some situations, when the phenomena appears very irregular, we solve this problem for jFj>0. In this case the providing an explanation which fits only part of it is better than nothing. Interestingly, Kearns et. al. did not consider the use of running time of our algorithm is bounded by a polynomial queries (but rather examples drawn from an arbitrary distribuand exponential ind. Our algorithm generalizes a previously tion) as they were skeptical that queries could be of any help. known algorithm, due to Goldreich and Levin, that solves this We show that queries do seem to help (see below). task for the case whenF=GF(2)(andd=1).
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 87 (5 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.
Extracting all the Randomness and Reducing the Error in Trevisan's Extractors
 In Proceedings of the 31st Annual ACM Symposium on Theory of Computing
, 1999
"... We give explicit constructions of extractors which work for a source of any minentropy on strings of length n. These extractors can extract any constant fraction of the minentropy using O(log² n) additional random bits, and can extract all the minentropy using O(log³ n) additional rando ..."
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Cited by 78 (16 self)
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We give explicit constructions of extractors which work for a source of any minentropy on strings of length n. These extractors can extract any constant fraction of the minentropy using O(log² n) additional random bits, and can extract all the minentropy using O(log³ n) additional random bits. Both of these constructions use fewer truly random bits than any previous construction which works for all minentropies and extracts a constant fraction of the minentropy. We then improve our second construction and show that we can reduce the entropy loss to 2 log(1=") +O(1) bits, while still using O(log³ n) truly random bits (where entropy loss is defined as [(source minentropy) + (# truly random bits used) (# output bits)], and " is the statistical difference from uniform achieved). This entropy loss is optimal up to a constant additive term. our...
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 77 (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)).
In Search of an Easy Witness: Exponential Time vs. Probabilistic Polynomial Time
"... Restricting the search space f0; 1g 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 cla ..."
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Cited by 55 (5 self)
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Restricting the search space f0; 1g 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 ae 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.