On Extractors and Exposure-Resilient Functions for Sublogarithmic Entropy

TL;DR

This paper improves explicit constructions of extractors for sources with very low min-entropy, showing optimal output length for certain classes and establishing probabilistic thresholds for exposure-resilient functions.

Contribution

It simplifies and enhances extractor constructions for sublogarithmic min-entropy sources and characterizes when random functions serve as effective extractors and exposure-resilient functions.

Findings

Error of extractors is exponentially small in k.

Optimal output length is achieved for sublogarithmic k.

Random functions serve as extractors when k is superlogarithmic in n.

Abstract

We study deterministic extractors for oblivious bit-fixing sources (a.k.a. resilient functions) and exposure-resilient functions with small min-entropy: of the function's n input bits, k << n bits are uniformly random and unknown to the adversary. We simplify and improve an explicit construction of extractors for bit-fixing sources with sublogarithmic k 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 an extractor for oblivious bit-fixing sources 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. log log n). No explicit exposure-resilient functions achieving these parameters are known.