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 can extract Ω(logk – logd – loglog(1/ε)) bits with error ε, for any n,d,k ∈ ℕ and ε∈(0,1). For sources with a larger min-entropy, we can extract even more randomness. When k≥n ^1/2 + γ, for any constant γ∈(0,1/2), we can extract m=k–O(d log(1/ε)) bits with any error $\varepsilon \ge 2^{-\Omega(n^{\gamma})}$. When k≥log^c n, for some constant c>0, we can extract m=k–d (1/ε)^O(1) bits with any error ε≥k ^{− − Ω(1)}. Our results generalize those of Kamp & 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 non-explicit deterministic extractor which can extract m=k–O(log(1/ε)) bits whenever k=ω(d+log(n/ε)). Finally, we show that even to extract from bit-fixing sources, any extractor, seeded or not, must suffer an entropy loss k–m = Ω(log(1/ε)). This generalizes a lower bound of Radhakrishnan & Ta-Shma with respect to general sources.