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

Ran Raz, Omer Reingold, Salil Vadhan

Research output: Contribution to journalArticle

70 Scopus citations

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 2 log(1/ε) + 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 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.

Original languageEnglish (US)
Pages (from-to)97-128
Number of pages32
JournalJournal of Computer and System Sciences
Volume65
Issue number1
DOIs
StatePublished - Aug 2002
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Networks and Communications
  • Computational Theory and Mathematics
  • Applied Mathematics

Keywords

  • Combinatorial designs
  • Expander graphs
  • Extractors
  • Probabilistic method
  • Pseudorandom generators

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