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 rst extracts any constant fraction of the min-entropy using O(log n) additional random bits. The second extracts all the min-entropy using O(log 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" su ces for the Nisan{Wigderson pseudorandom generator [NW94], which underlies the recent extractor of Trevisan [Tre98]. 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.