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 Olog2 n additional random bits. The second extracts all the min-entropy using Olog3 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.

• Type: Journal
• Year: 1998
• Where: ECCC
• Authors: Salil P. Vadhan
• Keywords: Additional Random Bits, Constant Fraction, ECCC 1998, Nisan Wigderson