NDLI: Extracting Randomness via Repeated Condensing

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Extracting Randomness via Repeated Condensing

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Wigderson, Avi Shaltiel, Ronen Reingold, Omer
Copyright Year	2006
Abstract	Extractors (as defined by Nisan and Zuckerman) are procedures that use a small number of truly random bits (called the seed) to extract many (almost) truly random bits from arbitrary distributions as long as distributions have sufficient (min)-entropy. A natural weakening of an extractor is a condenser, whose output distribution has a higher entropy rate than the input distribution (without losing much of the initial entropy). An extractor can be viewed as an ultimate condenser because it outputs a distribution with the maximal entropy rate.In this paper we construct explicit condensers with short seed length. The condenser constructions combine (variants of or more efficient versions of) ideas from several works, including the block extraction scheme of [N. Nisan and D. Zuckerman, J. Comput. System Sci., 52 (1996), pp. 43-52], the observation made in [A. Srinivasanand D. Zuckerman, SIAM J. Comput., 28 (1999), pp. 1433-1459; N. Nisan and A. Ta-Shma, J. Comput. System Sci., 58 (1999), pp. 148-173] that a failure of the block extraction scheme is also useful, the recursive "win-win" case analysis of [R. Impagliazzo, R. Shaltiel, and A. Wigderson, Near-optimal conversion of hardness into pseudo-randomness, in Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science, IEEE, Los Alamitos, CA, 1999, pp. 181-190; R. Impagliazzo, R. Shaltiel, and A. Wigderson, Extractors and pseudo-random generators with optimal seed length, in Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, ACM, New York, 2000, pp. 1-10], and the error correction of random sources used in [L. Trevisan, J. ACM, 48 (2001), pp. 860-879]. As a by-product (via repeated iterating of condensers), we obtain new extractor constructions. The new extractors give significant qualitative improvements over previous ones for sources of arbitrary min-entropy; they are nearly optimal simultaneously in the two main parameters of seed length and output length. Specifically, our extractors can make any one of these two parameters optimal (up to a constant factor) only at a polylogarithmic loss in the other. Previous constructions require polynomial loss in both cases for general sources.We also give a simple reduction converting "standard" extractors (which are good for an average seed) into "strong" ones (which are good for most seeds), with essentially the same parameters. With this reduction, all the above improvements apply to strong extractors as well.
Starting Page	1185
Ending Page	1209
Page Count	25
File Format	PDF
ISSN	00975397
DOI	10.1137/S0097539703431032
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	5
Volume Number	35
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-27
Access Restriction	Subscribed
Subject Keyword	randomness condensers derandomization Combinatorics randomness extractors
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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