Extracting randomness using few independent sources

Boaz Barak*, Russell Impagliazzo, Avi Wigderson

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

71 Scopus citations

Abstract

In this work we give the first deterministic extractors from a constant number of weak sources whose entropy rate is less than 1/2. Specifically, for every δ > 0 we give an explicit construction for extracting randomness from a constant (depending polynomially on 1/δ) number of distributions over {0,1}n each having min-entropy δn. These extractors output n bits that are 2-n close to uniform. This construction uses several results from additive number theory, and in particular a recent result of Bourgain et al. We also consider the related problem of constructing randomness dispersers. For any constant output length m, our dispersers use a constant number of identical distributions, each with δ-entropy ω(log n), and outputs every possible m-bit string with positive probability. The main tool we use is a variant of the "stepping-up lemma" of Erdos and Hajnal used in establishing a lower bound on the Ramsey number for hypergraphs.

Original languageEnglish
Pages (from-to)1095-1118
Number of pages24
JournalSIAM Journal on Computing
Volume36
Issue number4
DOIs
StatePublished - 2006
Externally publishedYes

Keywords

  • Ramsey graphs
  • Randomness extractors
  • Sum-product theorem

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