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Redondo Beach, California

Nov. 12, 2000 to Nov. 14, 2000

ISBN: 0-7695-0850-2

pp: 22

O. Reingold , AT&T Labs.-Res., Florham Park, NJ, USA

A. Wigderson , AT&T Labs.-Res., Florham Park, NJ, USA

ABSTRACT

On an input probability distribution with some (min-)entropy an extractor outputs a distribution with a (near) maximum entropy rate (namely the uniform distribution). A natural weakening of this concept is a condenser, whose output distribution has a higher entropy rate than the input distribution (without losing much of the initial entropy). We construct efficient explicit condensers. The condenser constructions combine (variants or more efficient versions of) ideas from several works, including the block extraction scheme of Nisan and Zuckerman (1996), the observation made by Srinivasan and Zuckerman (1994) and Nisan and Ta-Schma (1999) that a failure of the block extraction scheme is also useful, the recursive "win-win" case analysis of Impagliazzo et al. (1999, 2000), and the error correction of random sources used by Trevisan (1999). As a natural byproduct, (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 main two parameters-seed length and output length. Specifically, our extractors can make any of these two parameters optimal (up to a constant factor), only at a poly-logarithmic 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) to "strong " ones (which are good for mast seeds), with essentially the same parameters.

INDEX TERMS

probability; entropy; random processes; computational complexity; randomness extraction; repeated condensing; input probability distribution; entropy; maximum entropy rate; condenser; output distribution; block extraction scheme; recursive win-win case analysis; error correction; random sources; polynomial loss

CITATION

O. Reingold,
A. Wigderson,
"Extracting randomness via repeated condensing",

*FOCS*, 2000, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science 2000, pp. 22, doi:10.1109/SFCS.2000.892008