Computing with Very Weak Random Sources

We give an efficient algorithm to extract randomness from a very weak random source using a small additional number t of truly random bits. Our work extends that of Nisan and Zuckerman [ J. Comput. System Sci., 52 (1996), pp. 43--52] in that t remains small even if the entropy rate is well below constant. A key application of this is in running randomized algorithms using such a very weak source of randomness. For any fixed $\gamma 0$, we show how to simulate RP algorithms in time $n^{O(\log n)}$ using the output of a \ds\ with min-entropy $R^\gamma$. Such a weak random source is asked once for $R$ bits; it outputs an $R$-bit string according to any probability distribution that places probability at most $2^{-R^\gamma}$ on each string. If $\gamma 1/2$, our simulation also works for BPP; for $\gamma 1-1/(k+1)$, our simulation takes time $n^{O(\logk n)}$ (log(k) is the logarithm iterated k times). We also give a polynomial-time BPP simulation using Chor--Goldreich sources of min-entropy $R^{\Omega(1)}$, which is optimal. We present applications to time-space tradeoffs, expander constructions, and to the hardness of approximation. Of independent interest is our randomness-efficient Leftover Hash Lemma, a key tool for extracting randomness from weak random sources.