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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 80 (17 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...
Bucket Hashing and its Application to Fast Message Authentication
, 1995
"... We introduce a new technique for constructing a family of universal hash functions. ..."
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Cited by 51 (4 self)
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We introduce a new technique for constructing a family of universal hash functions.
Weak Random Sources, Hitting Sets, and BPP Simulations
, 1998
"... We show how to simulate any BPP algorithm in polynomial time using a weak random source of r bits and minentropy r fl for any fl ? 0. This follows from a more general result about sampling with weak random sources. Our result matches an informationtheoretic lower bound and solves a question that ..."
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Cited by 40 (5 self)
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We show how to simulate any BPP algorithm in polynomial time using a weak random source of r bits and minentropy r fl for any fl ? 0. This follows from a more general result about sampling with weak random sources. Our result matches an informationtheoretic 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 quasipolynomial time simulation of BPP [TaShma 1996]. Departing significantly from previous related works, we do not use extractors; instead, we use the ORdisperser 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...
Improved Randomness Extraction from Two Independent Sources
 In Proc. of 8th RANDOM
, 2004
"... Given two independent weak random sources X, Y , with the same length \ell and minentropies bX,bY whose sum is greater than \ell \Omega(polylog(\ell/\epsilon)), we construct a deterministic twosource extractor (aka "blender") that extracts max(bX,bY) (bX bY\ell4log(1/\epsilon)) bits which are \e ..."
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Cited by 23 (6 self)
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Given two independent weak random sources X, Y , with the same length \ell and minentropies bX,bY whose sum is greater than \ell \Omega(polylog(\ell/\epsilon)), we construct a deterministic twosource extractor (aka "blender") that extracts max(bX,bY) (bX bY\ell4log(1/\epsilon)) bits which are \epsilonclose to uniform. In contrast, best previously published construction [4] extracted at most 2(bX bY\ell2log(1/\epsilon)) bits. Our main technical tool is a construction of a strong twosource extractor that extracts (bX bY\ell2log(1/\epsilon)) bits which are \epsilonclose 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 twosource 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 twosource extractor constructions from the literature.
Extracting All the Randomness from a Weakly Random Source
, 1998
"... In this paper, we give two explicit constructions of extractors, both of which work for a source of any minentropy on strings of length n. The first extracts any constant fraction of the minentropy using O(log 2 n) additional random bits. The second extracts all the minentropy using O(log 3 ..."
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Cited by 6 (0 self)
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In this paper, we give two explicit constructions of extractors, both of which work for a source of any minentropy on strings of length n. The first extracts any constant fraction of the minentropy using O(log 2 n) additional random bits. The second extracts all the minentropy using O(log 3 n) additional random bits. Both constructions use fewer truly random bits than any previous construction which works for all minentropies and extracts a constant fraction of the minentropy. The extractors are obtained by observing that a weaker notion of "combinatorial design" suffices for the NisanWigderson pseudorandom generator [NW94], which underlies the recent extractor of Trevisan [Tre98]. We give nearoptimal constructions of such "weak designs" which achieve much better parameters than possible with the notion of designs used by NisanWigderson and Trevisan. 1 Introduction Roughly speaking, an extractor is a function which extracts truly random bits from a weakly random source,...