Extracting all the randomness and reducing the error in Trevisan's extractors

  • Authors:

  • Ran Raz;Omer Reingold;Salil Vadhan

  • Affiliations:

  • Department of Applied Mathematics and Computer Science, Weizmann Institute, Rehovot, 76100 Israel;AT&T Labs--Research, Building 103, 180 Park Avenue Florham Park, New Jersey;Division of Engineering and Applied Sciences, Harvard University, 33 Oxford Street, Cambridge, Massachusetts

  • Venue:
  • Journal of Computer and System Sciences - STOC 1999
  • Year:
  • 2002

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Abstract

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 2log(1/ε)+O(1) bits, while still using O(log3n) truly random bits (where entropy loss is defined as [(source min-entropy)+(# truly random bits used)-(# output bits)], and ε is the statistical difference from uniform achieved). This entropy loss is optimal up to a constant additive term. Our extractors are obtained by observing that a weaker notion of "combinatorial design" suffices for the Nisan-Wigderson pseudorandom generator, which underlies the recent extractor of Trevisan. 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. We also show how to improve our constructions (and Trevisan's construction) when the required statistical difference ε from the uniform distribution is relatively small. This improvement is obtained by using multilinear error-correcting codes over finite fields, rather than the arbitrary error-correcting codes used by Trevisan.