Loss-less condensers, unbalanced expanders, and extractors

A. Ta-Shma*, C. Umans, D. Zuckerman

*Corresponding author for this work

Research output: Contribution to journalConference articlepeer-review

Abstract

An extractor is a procedure which extracts randomness from a defective random source using a few additional random bits. Explicit extractor constructions have numerous applications and obtaining such constructions is an important de-randomization goal. Trevisan recently introduced an elegant extractor construction, but the number of truly random bits required is suboptimal when the input source has low minentropy. Significant progress toward overcoming this bottleneck has been made, but so far has required complicated recursive techniques that lose the simplicity of Trevisan's construction. We give a clean method for overcoming this bottleneck by constructing loss-less condensers, which compress the n-bit input source without losing any min-entropy, using O(log n) additional random bits. Our condensers are built using a simple modification of Trevisan's construction, and yield the best extractor constructions to date. Loss-less condensers also produce unbalanced bipartite expander graphs with small (poly logarithmic) degree D and very strong expansion of (1 - ε)D. We give other applications of our construction, including dispersers with entropy loss O(log n), depth two super-concentrators whose size is within a polylog of optimal, and an improved hardness of approximation result.

Original languageEnglish
Pages (from-to)143-152
Number of pages10
JournalConference Proceedings of the Annual ACM Symposium on Theory of Computing
DOIs
StatePublished - 2001
Event33rd Annual ACM Symposium on Theory of Computing - Creta, Greece
Duration: 6 Jul 20018 Jul 2001

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