Extracting All the Randomness from a Weakly Random Source

In this paper, we give two explicit constructions of extractors, both of which work for a source of any min-entropy on strings of length n. The first extracts any constant fraction of the min-entropy using O(log^2 n) additional random bits. The second extracts all the min-entropy using O(log^3 n) additional random bits Both 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. The 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.

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