On extractors and exposure‐resilient functions for sublogarithmic entropy1. Issue 3 (27th April 2012) - Reshef, Yakir; Vadhan, Salil
Abstract We study resilient functions and exposure‐resilient functions in the low‐entropy regime. A resilient function (a.k.a. deterministic extractor for oblivious bit‐fixing sources) maps any distribution on n ‐bit strings in which k bits are uniformly random and the rest are fixed into an output distribution that is close to uniform. With exposure‐resilient functions, all the input bits are random, but we ask that the output be close to uniform conditioned on any subset of n ‐ k input bits. In this paper, we focus on the case that k is sublogarithmic in n. We simplify and improve an explicit construction of resilient functions for k sublogarithmic in n due to Kamp and Zuckerman (SICOMP 2006), achieving error exponentially small in k rather than polynomially small in k. Our main result is that when k is sublogarithmic in n, the short output length of this construction (O(log k) output bits) is optimal for extractors computable by a large class of space‐bounded streaming algorithms. Next, we show that a random function is a resilient function with high probability if and only if k is superlogarithmic in n, suggesting that our main result may apply more generally. In contrast, we show that a random function is a static (resp. adaptive) exposure‐resilient function with high probability even if k is as small as a constant (resp. loglog n). No explicit exposure‐resilient functions achieving these parameters are known. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013