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We show that any randomized algorithm that runs in space S and time T and uses poly(S ) random bits can be simulated using only O(S ) random bits in space S and time T+poly(S). A deterministic simulation in space S follows. Of independent interest is our main technical tool: a procedure which extracts randomness from a defective random source using a small additional number of truly random bits. ] 1996 Academic Press, Inc.

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We show that modified versions of the linear congruentia1 generator and the shift register generator are provably good for amplifying the correctness of a probabilistic algorithm. More precisely, if T random bits are needed for a BPP algorithm to be correct with probability at least 2/3, then O(r + k z ) bits are needed to improve this probabilit,y to 1 -2 -k . We also present a different pseudo-random generator that is optimal, up to a constant fact,or, in this regard: it uses only O ( r + k ) bits to improve the probability to 1 -2 -k . This generator is based on random walks on expanders. Our results do not depend on any unproven assumptions.Rext we show that our modified versions of the shift register and linear congruential generators can be used to sample from distributions using, in the limit, the information-theoretic lower bound on random bits.

ABSTRACT:We present the first efficient oblivious sampler that uses an optimal number of random bits, up to an arbitrary constant factor bigger than 1. Specifically, for any ␣ ) 0, it Ž .Ž y 1 . Ž y 1 y 1 . uses 1 q ␣ m q log ␥ random bits to output ds poly ⑀ , log ␥ , m sample pointsx ⑀ G1y␥. Our proof is based on an improved extractor construction. An extractor is a procedure which takes as input the output of a defective random source and a small number of truly random bits, and outputs a nearly random string. We present the first optimal extractor, up to constant factors, for defective random sources with constant entropy rate. We give applications to constructive leader election and reducing randomness in interactive Ž .