Extracting randomness using few independent sources

Boaz Barak*, Russell Impagliazzo, Avi Wigderson

*Corresponding author for this work

Research output: Contribution to journalConference articlepeer-review

64 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. Specifrically, 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, which are 2 -n close to uniform. This construction uses several results from additive number theory, and in particular a recent one by Bourgain, Katz and Tao [BKT03] and of Konyagin [Kon03]. 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 min-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" used in establishing lower bound on the Ramsey number for hyper-graphs (Erdos and Hajnal, [GRS80]).

Original languageEnglish
Pages (from-to)384-393
Number of pages10
JournalProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
StatePublished - 2004
Externally publishedYes
EventProceedings - 45th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2004 - Rome, Italy
Duration: 17 Oct 200419 Oct 2004

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