Weak Random Sources, Hitting Sets, and BPP Simulations

We show how to simulate any BPP algorithm in polynomial time by using a weak random source of r bits and min-entropy $r^{\gamma}$ for any $\gamma >0$. 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 [M. Saks, A. Srinivasan, and S. Zhou, Proc. 27th ACM Symp. on Theory of Computing, 1995, pp. 479--488] and a quasi-polynomial time simulation of BPP [A. Ta-Shma, Proc. 28th ACM Symp. on Theory of Computing, 1996, pp. 276--285]. Departing significantly from previous related works, we do not use extractors; instead, we use the OR-disperser of Saks, Srinivasan, and Zhou in combination with a tricky use of hitting sets borrowed from [Andreev, Clementi, and Rolim, J. ACM, 45 (1998), pp. 179--213].