Extracting all the Randomness and Reducing the Error in Trevisan's Extractors
Ran Raz, Omer Reingold, and Salil Vadhan.
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.
Proceedings of 31st Annual ACM Symposium on Theory of Computing,
pp. 149-158, May 1999. ACM.
Electronic Colloquium on Computational Complexity Technical Report
Journal of Computer & System Sciences 65, pages 97-128,
Issue on STOC `99. [
compressed postscript ][official
back to Salil Vadhan's research interests ]