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35
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 43 (13 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 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 39 (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.
2source dispersers for subpolynomial entropy and Ramsey graphs beating the FranklWilson construction
 Proceedings of STOC06
, 2006
"... The main result of this paper is an explicit disperser for two independent sources on n bits, each of entropy k = n o(1). Put differently, setting N = 2 n and K = 2 k, we construct explicit N × N Boolean matrices for which no K × K submatrix is monochromatic. Viewed as adjacency matrices of bipartit ..."
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Cited by 27 (6 self)
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The main result of this paper is an explicit disperser for two independent sources on n bits, each of entropy k = n o(1). Put differently, setting N = 2 n and K = 2 k, we construct explicit N × N Boolean matrices for which no K × K submatrix is monochromatic. Viewed as adjacency matrices of bipartite graphs, this gives an explicit construction of KRamsey bipartite graphs of size N. This greatly improves the previous bound of k = o(n) of Barak, Kindler, Shaltiel, Sudakov and Wigderson [4]. It also significantly improves the 25year record of k = Õ( √ n) on the special case of Ramsey graphs, due to Frankl and Wilson [9]. The construction uses (besides ”classical ” extractor ideas) almost all of the machinery developed in the last couple of years for extraction from independent sources, including: • Bourgain’s extractor for 2 independent sources of some entropy rate < 1/2 [5] • Raz’s extractor for 2 independent sources, one of which has any entropy rate> 1/2 [18] • Rao’s extractor for 2 independent blocksources of entropy n Ω(1) [17]
Extracting Kolmogorov complexity with applications to dimension zeroone laws
 IN PROCEEDINGS OF THE 33RD INTERNATIONAL COLLOQUIUM ON AUTOMATA, LANGUAGES, AND PROGRAMMING
, 2006
"... We apply recent results on extracting randomness from independent sources to "extract " Kolmogorov complexity. For any ff; ffl? 0, given a string x with K(x) ? ffjxj, we show how to use a constant number of advice bits to efficiently compute another string y, jyj = \Omega (jxj), with K(y ..."
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Cited by 20 (4 self)
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We apply recent results on extracting randomness from independent sources to "extract " Kolmogorov complexity. For any ff; ffl? 0, given a string x with K(x) ? ffjxj, we show how to use a constant number of advice bits to efficiently compute another string y, jyj = \Omega (jxj), with K(y) ? (1 \Gamma ffl)jyj. This result holds for both classical and spacebounded Kolmogorov complexity. We use the extraction procedure for spacebounded complexity to establish zeroone laws for polynomialspace strong dimension. Our results include: (i) If Dimpspace(E) ? 0, then Dimpspace(E=O(1)) = 1. (ii) Dim(E=O(1) j ESPACE) is either 0 or 1. (iii) Dim(E=poly j ESPACE) is either 0 or 1. In other words,
An exposition of bourgain’s 2source extractor
, 2007
"... A construction of Bourgain [Bou05] gave the first 2source extractor to break the minentropy rate 1/2 barrier. In this note, we write an exposition of his result, giving a high level way to view his extractor construction. We also include a proof of a generalization of Vazirani’s XOR lemma that see ..."
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Cited by 14 (0 self)
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A construction of Bourgain [Bou05] gave the first 2source extractor to break the minentropy rate 1/2 barrier. In this note, we write an exposition of his result, giving a high level way to view his extractor construction. We also include a proof of a generalization of Vazirani’s XOR lemma that seems interesting in its own right, and an argument (due to Boaz Barak) that shows that any two source extractor with sufficiently small error must be strong.
Extractors and rank extractors for polynomial sources
 In FOCS ’07
, 2007
"... In this paper we construct explicit deterministic extractors from polynomial sources, which are distributions sampled by low degree multivariate polynomials over finite fields. This naturally generalizes previous work on extraction from affine sources (which are degree 1 polynomials). A direct conse ..."
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Cited by 10 (6 self)
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In this paper we construct explicit deterministic extractors from polynomial sources, which are distributions sampled by low degree multivariate polynomials over finite fields. This naturally generalizes previous work on extraction from affine sources (which are degree 1 polynomials). A direct consequence is a deterministic extractor for distributions sampled by polynomial size arithmetic circuits over exponentially large fields. The steps in our extractor construction, and the tools (mainly from algebraic geometry) that we use for them, are of independent interest: The first step is a construction of rank extractors, which are polynomial mappings which ”extract” the algebraic rank from any system of low degree polynomials. More precisely, for any n polynomials, k of which are algebraically independent, a rank extractor outputs k algebraically independent polynomials of slightly higher degree. The rank extractors we construct are applicable not only over finite fields but also over fields of characteristic zero. The next step is relating algebraic independence to minentropy. We use a theorem of Wooley to show that these parameters are tightly connected. This allows replacing the algebraic assumption on the source (above) by the natural information theoretic one. It also shows that a rank extractor is already a high quality condenser for polynomial sources over polynomially large fields. Finally, to turn the condensers into extractors, we employ a theorem of Bombieri, giving a character sum estimate for polynomials defined over curves. It allows extracting all the randomness (up to a multiplicative constant) from polynomial sources over exponentially large prime fields.
Multilinear Formulas, MaximalPartition Discrepancy and MixedSources Extractors
, 2007
"... We study a new method for proving lower bounds for subclasses of arithmetic circuits. Roughly speaking, the lower bound is proved by bounding the correlation between the coefficients ’ vector of a polynomial and the coefficients ’ vector of any product of two polynomials with disjoint sets of variab ..."
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Cited by 9 (5 self)
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We study a new method for proving lower bounds for subclasses of arithmetic circuits. Roughly speaking, the lower bound is proved by bounding the correlation between the coefficients ’ vector of a polynomial and the coefficients ’ vector of any product of two polynomials with disjoint sets of variables. We prove lower bounds for several old and new subclasses of circuits. Monotone Circuits: We prove a tight 2 Ω(n) lower bound for the size of monotone arithmetic circuits. The highest previous lower bound was 2 Ω( √ n). Orthogonal Formulas: We prove a tight 2 Ω(n) lower bound for the size of orthogonal multilinear formulas (defined, motivated, and studied by Aaronson). Previously, nontrivial lower bounds were only known for subclasses of orthogonal multilinear formulas. NonCancelling Formulas: We define and study the new model of noncancelling multilinear formulas. Roughly speaking, in this model one is not allowed to sum two polynomials that almost cancel each other. The noncancelling multilinear model is a generalization of both the monotone model and the orthogonal model. We prove lower bounds of n Ω(1) for the depth of noncancelling multilinear formulas.
2Source Extractors Under Computational Assumptions and Cryptography with Defective Randomness
"... Abstract — We show how to efficiently extract truly random bits from two independent sources of linear minentropy, under a computational assumption. The assumption we rely on is the existence of an efficiently computable permutation f 1, such that for any source X ∈{0, 1} n with linear minentropy, ..."
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Cited by 8 (1 self)
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Abstract — We show how to efficiently extract truly random bits from two independent sources of linear minentropy, under a computational assumption. The assumption we rely on is the existence of an efficiently computable permutation f 1, such that for any source X ∈{0, 1} n with linear minentropy, any circuit of size poly(n) cannot invert f(X) with nonnegligible probability. Under the stronger assumption that f(X) cannot be inverted even by circuits of size poly(n logn) with nonnegligible probability, we design a lossless computational network extractor protocol. Namely, we design a protocol for a set of players, each with access to an independent source of linear minentropy, with the guarantee that at the end of the protocol, each honest player is left with bits that are computationally indistinguishable from being uniform and private. Our protocol succeeds as long as there are at least two honest players. Our results imply that if such oneway permutations exist, and enhanced trapdoor permutations exist, then secure multiparty computation with imperfect randomness is possible for any number of players, as long as at least two of them are honest. We also construct a network extractor protocol for the case where each source has only polynomiallysmall minentropy (n δ for some constant δ> 0). For this we need at least a constant u(δ) (which depends on δ) number of honest players, and we need that the oneway permutation is hard to invert even on polynomially small minentropy sources. 1.
Increasing the Output Length of ZeroError Dispersers
, 2008
"... Let C be a class of probability distributions over a finite set Ω. A function D: Ω ↦ → {0, 1} m is a disperser for C with entropy threshold k and error ɛ if for any distribution X in C such that X gives positive probability to at least 2k elements we have that the distribution D(X) gives positive pr ..."
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Cited by 6 (5 self)
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Let C be a class of probability distributions over a finite set Ω. A function D: Ω ↦ → {0, 1} m is a disperser for C with entropy threshold k and error ɛ if for any distribution X in C such that X gives positive probability to at least 2k elements we have that the distribution D(X) gives positive probability to at least (1 − ɛ)2m elements. A long line of research is devoted to giving explicit (that is polynomial time computable) dispersers (and related objects called “extractors”) for various classes of distributions while trying to maximize m as a function of k. In this paper we are interested in explicitly constructing zeroerror dispersers (that is dispersers with error ɛ = 0). For several interesting classes of distributions there are explicit constructions in the literature of zeroerror dispersers with “small ” output length m and we give improved constructions that achieve “large ” output length, namely m = Ω(k). We achieve this by developing a general technique to improve the output length of zeroerror dispersers (namely, to transform a disperser with short output length into one with large output length). This strategy works for several classes of sources and is inspired by a transformation that improves the output length of extractors (which was given in [31] building on earlier work