Sparse extractor families for all the entropy

TL;DR

This paper introduces sparse extractor families capable of extracting nearly all entropy from distributions with minimal assumptions, providing tight bounds on their sparsity and applications in pseudorandom generator construction.

Contribution

The paper presents new constructions of sparse extractor families that work for all high min-entropy distributions without distributional assumptions, with tight bounds on sparsity.

Findings

Tight bounds on sparsity for strong extractor families across various min-entropies

Existence of weak extractor families with better sparsity for certain min-entropies

Application of sparse extractors in efficient parallel transformation of one-way functions

Abstract

We consider the problem of extracting entropy by sparse transformations, namely functions with a small number of overall input-output dependencies. In contrast to previous works, we seek extractors for essentially all the entropy without any assumption on the underlying distribution beyond a min-entropy requirement. We give two simple constructions of sparse extractor families, which are collections of sparse functions such that for any distribution X on inputs of sufficiently high min-entropy, the output of most functions from the collection on a random input chosen from X is statistically close to uniform. For strong extractor families (i.e., functions in the family do not take additional randomness) we give upper and lower bounds on the sparsity that are tight up to a constant factor for a wide range of min-entropies. We then prove that for some min-entropies weak extractor families can achieve better sparsity. We show how this construction can be used towards more efficient parallel transformation of (non-uniform) one-way functions into pseudorandom generators. More generally, sparse extractor families can be used instead of pairwise independence in various randomized or nonuniform settings where preserving locality (i.e., parallelism) is of interest.