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63
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.
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)).
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 51 (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.
Linear degree extractors and the inapproximability of max clique and chromatic number
 THEORY OF COMPUTING
, 2007
"... ... that for all ε> 0, approximating MAX CLIQUE and CHROMATIC NUMBER to within n1−ε are NPhard. We further derandomize results of Khot (FOCS ’01) and show that for some γ> 0, no quasipolynomial time algorithm approximates MAX CLIQUE or CHROMATIC NUMBER to within n/2 (logn)1−γ, unless N˜P = ˜P. The ..."
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Cited by 46 (0 self)
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... that for all ε> 0, approximating MAX CLIQUE and CHROMATIC NUMBER to within n1−ε are NPhard. We further derandomize results of Khot (FOCS ’01) and show that for some γ> 0, no quasipolynomial time algorithm approximates MAX CLIQUE or CHROMATIC NUMBER to within n/2 (logn)1−γ, unless N˜P = ˜P. The key to these results is a new construction of dispersers, which are related to randomness extractors. A randomness extractor is an algorithm which extracts randomness from a lowquality random source, using some additional truly random bits. We construct new extractors which require only log2 n + O(1) additional random bits for sources with constant entropy rate, and have constant error. Our dispersers use an arbitrarily small constant
Extracting Randomness via Repeated Condensing
 In Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science
, 2000
"... On an input probability distribution with some (min)entropy an extractor outputs a distribution with a (near) maximum entropy rate (namely the uniform distribution). A natural weakening of this concept is a condenser, whose output distribution has a higher entropy rate than the input distribution ( ..."
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Cited by 43 (16 self)
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On an input probability distribution with some (min)entropy an extractor outputs a distribution with a (near) maximum entropy rate (namely the uniform distribution). A natural weakening of this concept is a condenser, whose output distribution has a higher entropy rate than the input distribution (without losing much of the initial entropy). In this paper we construct efficient explicit condensers. The condenser constructions combine (variants or more efficient versions of) ideas from several works, including the block extraction scheme of [NZ96], the observation made in [SZ94, NT99] that a failure of the block extraction scheme is also useful, the recursive "winwin" case analysis of [ISW99, ISW00], and the error correction of random sources used in [Tre99]. As a natural byproduct, (via repeated iterating of condensers), we obtain new extractor constructions. The new extractors give significant qualitative improvements over previous ones for sources of arbitrary minentropy; they...
Simulating Independence: New Constructions of Condensers, Ramsey Graphs, Dispersers, and Extractors
 In Proceedings of the 37th Annual ACM Symposium on Theory of Computing
, 2005
"... We present new explicit constructions of deterministic randomness extractors, dispersers and related objects. More precisely, a distribution X over binary strings of length n is called a δsource if it assigns probability at most 2 −δn to any string of length n, and for any δ> 0 we construct the fol ..."
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Cited by 41 (12 self)
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We present new explicit constructions of deterministic randomness extractors, dispersers and related objects. More precisely, a distribution X over binary strings of length n is called a δsource if it assigns probability at most 2 −δn to any string of length n, and for any δ> 0 we construct the following poly(n)time computable functions: 2source disperser: D: ({0, 1} n) 2 → {0, 1} such that for any two independent δsources X1, X2 we have that the support of D(X1, X2) is {0, 1}. Bipartite Ramsey graph: Let N = 2 n. A corollary is that the function D is a 2coloring of the edges of KN,N (the complete bipartite graph over two sets of N vertices) such that any induced subgraph of size N δ by N δ is not monochromatic. 3source extractor: E: ({0, 1} n) 2 → {0, 1} such that for any three independent δsources X1, X2, X3 we have that E(X1, X2, X3) is (o(1)close to being) an unbiased random bit. No previous explicit construction was known for either of these, for any δ < 1/2 and these results constitute major progress to longstanding open problems. A component in these results is a new construction of condensers that may be of independent
Extractors from ReedMuller Codes
 In Proceedings of the 42nd Annual IEEE Symposium on Foundations of Computer Science
, 2001
"... Finding explicit extractors is an important derandomization goal that has received a lot of attention in the past decade. This research has focused on two approaches, one related to hashing and the other to pseudorandom generators. A third view, regarding extractors as good error correcting codes, w ..."
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Cited by 39 (5 self)
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Finding explicit extractors is an important derandomization goal that has received a lot of attention in the past decade. This research has focused on two approaches, one related to hashing and the other to pseudorandom generators. A third view, regarding extractors as good error correcting codes, was noticed before. Yet, researchers had failed to build extractors directly from a good code, without using other tools from pseudorandomness. We succeed in constructing an extractor directly from a ReedMuller code. To do this, we develop a novel proof technique. Furthermore, our construction is the first and only construction with degree close to linear. In contrast, the best previous constructions had brought the log of the degree within a constant of optimal, which gives polynomial degree. This improvement is important for certain applications. For example, it follows that approximating the VC dimension to within a factor of N
Extractors for a constant number of polynomially small minentropy independent sources
 In Proceedings of the 38th Annual ACM Symposium on Theory of Computing
, 2006
"... We consider the problem of randomness extraction from independent sources. We construct an extractor that can extract from a constant number of independent sources of length n, each of which have minentropy n γ for an arbitrarily small constant γ> 0. Our extractor is obtained by composing seeded ex ..."
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Cited by 38 (10 self)
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We consider the problem of randomness extraction from independent sources. We construct an extractor that can extract from a constant number of independent sources of length n, each of which have minentropy n γ for an arbitrarily small constant γ> 0. Our extractor is obtained by composing seeded extractors in simple ways. We introduce a new technique to condense independent somewhererandom sources which looks like a useful way to manipulate independent sources. Our techniques are different from those used in recent work [BIW04, BKS + 05, Raz05, Bou05] for this problem in the sense that they do not rely on any results from additive number theory. Using Bourgain’s extractor [Bou05] as a black box, we obtain a new extractor for 2 independent blocksources with few blocks, even when the minentropy is as small as polylog(n). We also show how to modify the 2 source disperser for linear minentropy of Barak et al. [BKS + 05] and the 3 source extractor of Raz [Raz05] to get dispersers/extractors with exponentially small error and linear output length where previously both were constant. In terms of Ramsey Hypergraphs, for every constant 1> γ> 0 our construction gives a family of explicit O(1/γ)uniform hypergraphs on N vertices that avoid cliques and independent sets of (log N)γ size 2.
Time of deterministic broadcasting in radio networks with local knowledge
 SIAM Journal on Computing
, 2004
"... Abstract. We consider broadcasting in radio networks, modeled as undirected graphs, whose nodes know only their own label and labels of their neighbors. In every step every node acts either as a transmitter or as a receiver. A node acting as a transmitter sends a message which can potentially reach ..."
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Cited by 30 (8 self)
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Abstract. We consider broadcasting in radio networks, modeled as undirected graphs, whose nodes know only their own label and labels of their neighbors. In every step every node acts either as a transmitter or as a receiver. A node acting as a transmitter sends a message which can potentially reach all of its neighbors. A node acting as a receiver in a given step gets a message if and only if exactly one of its neighbors transmits in this step. BarYehuda, Goldreich, and Itai [J. Comput. System Sci., 45 (1992), pp. 104–126] considered broadcasting in this model. They claimed a linear lower bound on the time of deterministic broadcasting in such radio networks of diameter 3. This claim turns out to be incorrect in this model (although it is valid in a more pessimistic model [R. BarYehuda, O. Goldreich, and A. Itai, Errata Regarding “On the time complexity of broadcast in radio networks: An exponential gap between determinism and randomization, ”