Deterministic extractors for independent-symbol sources

• Authors: Chia-Jung Lee;Chi-Jen Lu;Shi-Chun Tsai
• Venue: IEEE Transactions on Information Theory
• Year: 2010

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 Ω(log k- log log(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 ≥ n1/2+γ, for any constant γ ∈ (0, 1/2), we can extract m=k-O(d log(1/ε)) bits with any error ε ≥ 2-Ω(nγ). When k ≥ logc n, for some constant c 0, we can extract m=k-(1/ε)O(1) bits with any error ε ≥ k-Ω(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/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 and Ta-Shma on extracting from general sources.