Deterministic extractors for independent-symbol sources

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(log k -log d-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 ≥ 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 ε ≥ 2 ^Ωnγ, 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 = w(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.