Deterministic Extractors for Independent-Symbol Sources

Abstract

In this paper, we consider the task of deterministically extracting randomness from sources consisting of a sequence of n independent symbols from {0, 1}(d). The only randomness guarantee on such a source is that the whole source has min-entropy k. We give an explicit deterministic extractor which extract Omega(log k - log log(1/epsilon)) bits with error epsilon, for any, n, d, k is an element of N and epsilon is an element of (0, 1). For sources with a larger min-entropy, we can extract even more randomness. When k >= n(1/2+gamma), for any constant gamma is an element of (1, 1/2), we can extract m = k - O(d log(1/epsilon)) bits with any error epsilon >= 2(-Omega(n gamma)). When k >= log(c) n, for some constant c > 0 , we can extract m = k - (1/epsilon)(O(1)) bits with any error epsilon >= k(-Omega(1)). Our results generalize those of Kamp and Zuckerman and Gabizon et al. which only work for bit-fixing sources (with d = 1 and each bit of the source being either fixed or perfectly random). Moreover, we show the existence of a nonexplicit deterministic extractor which can extract m = k - O(log(1/epsilon)) bits whenever k = omega(d + log(n/epsilon)). Finally, we show that even to extract from bit-fixing sources, any extractor, seeded or not, must suffer an entropy loss k - m = Omega(log(1/epsilon)). This generalizes a lower bound of Radhakrishnan and Ta-Shma on extracting from general sources.