Zero-Fixing Extractors for Sub-Logarithmic Entropy

Abstract

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 $(1-o(1)) \cdot k$ bits for $k = \mathrm{poly}\log {n}$, 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 $n > \mathsf {Tower}(k^2)$ 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 $\Omega (k)$ bits for $k \ge \mathrm{poly}\log \log {n}$. Finally, we give a construction of an (n, k)-bit-fixing extractor, that outputs $k-O(1)$ bits, for entropy $k = (1+o(1)) \cdot \log \log {n}$, with running-time $n^{O(( \log {\log {n}})^2)}$. This answers an open problem by Reshef and Vadhan [RSA 2013].

G. Cohen—Supported by an ISF grant and by the I-CORE Program of the Planning and Budgeting Committee. I. Shinkar—Research supported by NSF grants CCF 1422159, 1061938, 0832795 and Simons Collaboration on Algorithms and Geometry grant.