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A “randomness extractor” is an algorithm that given a sample from a distribution with sufficiently high min-entropy and a short random seed produces an output that is statistically indistinguishable from uniform. (Min-entropy is a measure of the amount of randomness in a distribution.) We present a simple, self-contained extractor construction that produces good extractors for all min-entropies. Our construction is algebraic and builds on a new polynomial-based approach introduced by Ta-Shma et al. [2001b]. 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 min-entropy 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 Nisan-Wigderson generator [Nisan and Wigderson 1994], and turns worst-case hardness directly into pseudorandomness. The parameters of our generator match those in Impagliazzo and Wigderson [1997] and Sudan et al. [2001] and in particular are strong enough to obtain a new proof that P = BPP if E requires exponential size circuits.Our construction also gives the following improvements over previous work:---We construct an optimal “hitting set generator” that stretches O (log n ) random bits into s Ω(1) pseudorandom bits when given a function on log n bits that requires circuits of size s . This yields a quantitatively optimal hardness versus randomness tradeoff for both RP and BPP and solves an open problem raised in Impagliazzo et al. [1999].---We give the first construction of pseudorandom generators that fool nondeterministic circuits when given a function that requires large nondeterministic circuits. This technique also give a quantitatively optimal hardness versus randomness tradeoff for AM and the first hardness amplification result for nondeterministic circuits.

The main result of this paper is an explicit disperser for two independent sources on n bits, each of min-entropy k = 2 log β n , where β < 1 is some absolute constant. Put differently, setting N = 2 n and K = 2 k , we construct an explicit N ×N Boolean matrix for which no K ×K sub-matrix is monochromatic. Viewed as the adjacency matrix of a bipartite graph, this gives an explicit construction of a bipartite K-Ramsey graph of 2N vertices. This improves the previous bound of k = o(n) of Barak, Kindler, Shaltiel, Sudakov and Wigderson. As a corollary, we get a construction of a 2 We also give a construction of a new independent sources extractor that can extract from a constant number of sources of polynomially small minentropy with exponentially small error. This improves independent sources extractor of Rao, which only achieved polynomially small error.Our dispersers combine ideas and constructions from several previous works in the area together with some new ideas. In particular, we rely on the extractors of Raz and Bourgain as well as an improved version of the extractor of Rao. A key ingredient that allows us to beat the barrier of k = √ n is a new and more complicated variant of the challenge-response mechanism of Barak et al. that allows us to locate the min-entropy concentrations in a source of low min-entropy.

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