Lossless Condensers, Unbalanced Expanders, And Extractors

  • Authors:

  • Amnon Ta-Shma;Christopher Umans†;David Zuckerman‡

  • Affiliations:

  • Tel-Aviv University, Computer Science Department, 69978, Tel-Aviv, Israel;California Institute of Technology, Computer Science, 1200 E. California Blvd, CA91125, Pasadena, USA;University of Texas, Department of Computer Science, 1 University Station C0500, TX 78712, Austin, USA

  • Venue:
  • Combinatorica
  • Year:
  • 2007

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Abstract

Trevisan showed that many pseudorandom generator constructions give rise to constructions of explicit extractors. We show how to use such constructions to obtain explicit lossless condensers. A lossless condenser is a probabilistic map using only O(logn) additional random bits that maps n bits strings to poly(logK) bit strings, such that any source with support size K is mapped almost injectively to the smaller domain. Our construction remains the best lossless condenser to date. By composing our condenser with previous extractors, we obtain new, improved extractors. For small enough min-entropies our extractors can output all of the randomness with only O(logn) bits. We also obtain a new disperser that works for every entropy loss, uses an O(logn) bit seed, and has only O(logn) entropy loss. This is the best disperser construction to date, and yields other applications. Finally, our lossless condenser can be viewed as an unbalanced bipartite graph with strong expansion properties.