We give explicit constructions of extractors which work for a source of any min-entropy on strings of length n. These extractors can extract any constant fraction of the min-entropy using O(log² n) additional random bits, and can extract all the min-entropy using O(log³ n) additional random bits. Both of these constructions use fewer truly random bits than any previous construction which works for all min-entropies and extracts a constant fraction of the min-entropy. 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 min-entropy) + (# 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...

We introduce a new technique for constructing a family of universal hash functions.

We show how to simulate any BPP algorithm in polynomial time using a weak random source of r bits and min-entropy r fl for any fl ? 0. This follows from a more general result about sampling with weak random sources. Our result matches an information-theoretic lower bound and solves a question that has been open for some years. The previous best results were a polynomial time simulation of RP [Saks, Srinivasan and Zhou 1995] and a quasi-polynomial time simulation of BPP [Ta-Shma 1996]. Departing significantly from previous related works, we do not use extractors; instead, we use the OR-disperser of [Saks, Srinivasan, and Zhou 1995] in combination with a tricky use of hitting sets borrowed from [Andreev, Clementi, and Rolim 1996]. AMS Subject Classification: 68Q10, 11K45. Key Words and Phrases: Derandomization, Imperfect Sources of Randomness, Hitting Sets, Randomized Computations, Expander Graphs. Abbreviated Title: BPP Simulations using Weak Random Sources. 1 Introduction Randomi...

Given two independent weak random sources X, Y , with the same length \ell and min-entropies bX,bY whose sum is greater than \ell \Omega(polylog(\ell/\epsilon)), we construct a deterministic two-source extractor (aka "blender") that extracts max(bX,bY) (bX bY-\ell-4log(1/\epsilon)) bits which are \epsilon-close to uniform. In contrast, best previously published construction [4] extracted at most 2(bX bY--\ell-2log(1/\epsilon)) bits. Our main technical tool is a construction of a strong two-source extractor that extracts (bX bY-\ell-2log(1/\epsilon)) bits which are \epsilon-close to being uniform and independent of one of the sources(aka "strong blender"), so that they can later be reused as a seed to a seeded extractor. Our strong two-source extractor construction improves the best previously published construction of such strong blenders [7] by a factor of 2, applies to more sources X and Y , and is considerably simpler than the latter. Our methodology also unifies several of the previous two-source extractor constructions from the literature.

In this paper, we give two explicit constructions of extractors, both of which work for a source of any min-entropy on strings of length n. The first extracts any constant fraction of the min-entropy using O(log 2 n) additional random bits. The second extracts all the min-entropy using O(log 3 n) additional random bits. Both constructions use fewer truly random bits than any previous construction which works for all min-entropies and extracts a constant fraction of the min-entropy. The extractors are obtained by observing that a weaker notion of "combinatorial design" suffices for the Nisan--Wigderson pseudorandom generator [NW94], which underlies the recent extractor of Trevisan [Tre98]. We give near-optimal constructions of such "weak designs" which achieve much better parameters than possible with the notion of designs used by Nisan--Wigderson and Trevisan. 1 Introduction Roughly speaking, an extractor is a function which extracts truly random bits from a weakly random source,...