TR99-046 Authors: Ran Raz, Omer Reingold, Salil Vadhan

Publication: 23rd November 1999 14:39

Downloads: 1324

Keywords:

We give explicit constructions of extractors which work for a source of

any min-entropy on strings of length n. These extractors can extract any

constant fraction of the min-entropy using O(log^2 n) additional random

bits, and can extract all the min-entropy using O(log^3 n) additional

random bits. Both of these constructions use fewer truly random bits than

any previous construction which works for all min-entropies and extracts a

constant fraction of the min-entropy. We then improve our second

construction and show that we can reduce the entropy loss to

2log(1/epsilon)+O(1) bits, while still using O(log^3 n) truly random bits

(where entropy loss is defined as [(source min-entropy)+ (# truly random

bits used)- (# output bits)], and epsilon is the statistical difference

from uniform achieved). This entropy loss is optimal up to a constant

additive term.

Our extractors are obtained by observing that a weaker notion of

"combinatorial design" suffices for the Nisan-Wigderson pseudorandom

generator, which underlies the recent extractor of Trevisan. We give

near-optimal constructions of such "weak designs" which achieve much

better parameters than possible with the notion of designs used by

Nisan-Wigderson and Trevisan.

We also show how to improve our constructions (and Trevisan's

construction) when the required statistical difference epsilon from the

uniform distribution is relatively small. This improvement is obtained by

using multilinear error-correcting codes over finite fields, rather than

the arbitrary error-correcting codes used by Trevisan.