Weak random sources, hitting sets, and BPP simulations

We show how to simulate any BPP algorithm in polynomial time using a weak random source of min-entropy r^γ for any γ>0. This follows from a more general result about sampling with weak random sources. Our result matches an information-theoretic lower bound and solves a question that has been open for some years. The previous best results were a polynomial time simulation of RP (Saks et al., 1995) and a n(log^(k)n)-time simulation of BPP for fixed k (Ta-Shma, 1996). Departing significantly from previous related works, we do not use extractors; instead we use the OR-disperser of (Saks et al., 1995) in combination with a tricky use of hitting sets borrowed from Andreev et al. (1996). Of independent interest is our new (simplified) proof of the main result of Andreev et al., (1996). Our proof also gives some new hardness/randomness trade-offs for parallel classes