Zero-Fixing Extractors for Sub-Logarithmic Entropy

An (n, k)-bit-fixing source is a distribution on n bit strings, that is fixed on n − k of the coordinates, and jointly uniform on the remaining k bits. Explicit constructions of bit-fixing extractors by Gabizon, Raz and Shaltiel [SICOMP 2006] and Rao [CCC 2009], extract (1 − o(1)) · k bits for k = poly log n, almost matching the probabilistic argument. Intriguingly, unlike other well-studied sources of randomness, a result of Kamp and Zuckerman [SICOMP 2006] shows that, for any k, some small portion of the entropy in an (n, k)-bit-fixing source can be extracted. Although the extractor does not extract all the entropy, it does extract (1/2− o(1)) · log(k) bits. In this paper we prove that when the entropy k is small enough compared to n, this exponential entropy-loss is unavoidable. More precisely, one cannot extract more than log(k)/2 + O(1) bits from (n, k)-bit-fixing sources. The remaining entropy is inaccessible, information theoretically. By the Kamp-Zuckerman construction, this negative result is tight. Our impossibility result also holds for what we call zero-fixing sources. These are bit-fixing sources where the fixed bits are set to 0. We complement our negative result, by giving an explicit construction of an (n, k)-zero-fixing extractor, that outputs Ω(k) bits, even for k = poly log log n. Furthermore, we give a construction of an (n, k)-bitfixing extractor, that outputs k −O(1) bits, for entropy k = (1 + o(1)) · log logn, with running-time nO((log logn) 2). ∗Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel. Email: [email protected]. †Courant Institute of Mathematical Sciences, New York University. Email: [email protected] ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 160 (2014)