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 log(k)/2 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, we show that for n > Tower(k 2) 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. For small enough k, this strengthens a result by Reshef and Vadhan [RSA 2013], who proved a similar bound for extractors computable by space-bounded streaming algorithms. 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 O(k)bits for k >= poly log log n. Finally, we give a construction of an (n, k)-bit-fixing extractor, that outputs k O( 1) bits, for entropy k = (1 + o(1)) . log log n, with running-time n O((log log n) 2). This answers an open problem by Reshef and Vadhan [RSA 2013].