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New Extractors for Interleaved Sources

Authors Eshan Chattopadhyay, David Zuckerman

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  • extractor
  • derandomization
  • explicit construction

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Abstract

We study how to extract randomness from a C-interleaved source, that is, a source comprised of C independent sources whose bits or symbols are interleaved. We describe a simple approach for constructing such extractors that yields:

(1) For some delta>0, c>0, explicit extractors for 2-interleaved sources on {0,1}^{2n} when one source has min-entropy at least (1-delta)*n and the other has min-entropy at least c*log(n). The best previous construction, by Raz and Yehudayoff, worked only when both sources had entropy rate 1-delta. 

(2) For some c>0 and any large enough prime p, explicit extractors for 2-interleaved sources on [p]^{2n} when one source has min-entropy rate at least .51 and the other source has min-entropy rate at least (c*log(n))/n.

We use these to obtain the following applications:

(a) We introduce the class of any-order-small-space sources, generalizing the class of small-space sources studied by Kamp et al.. We construct extractors for such sources with min-entropy rate close to 1/2. Using the Raz-Yehudayoff construction would require entropy rate close to 1.

(b) For any large enough prime p, we exhibit an explicit function f:[p]^{2n} -> {0,1} such that the randomized best-partition communication complexity of f with error 1/2-2^{-Omega(n)} is at least .24*n*log(p). Previously this was known only for a tiny constant instead of .24, for p=2 by by Raz and Yehudayoff.

We introduce non-malleable extractors in the interleaved model. For any large enough prime p, we give an explicit construction of a weak-seeded non-malleable extractor for sources over [p]^n with min-entropy rate .51. Nothing was known previously, even for almost full min-entropy.

Cite As Get BibTex

Eshan Chattopadhyay and David Zuckerman. New Extractors for Interleaved Sources. In 31st Conference on Computational Complexity (CCC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 50, pp. 7:1-7:28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.CCC.2016.7

Author Details

Eshan Chattopadhyay
David Zuckerman

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