Extractors for Polynomial Sources over $\mathbb{F}_2$

TL;DR

This paper presents the first explicit construction of extractors for polynomial sources over _2 with high min-entropy, introduces a novel input reduction lemma, and demonstrates inherent limitations of sumset extractors for such sources.

Contribution

It provides the first nontrivial extractor for polynomial sources over _2 and introduces polynomial NOBF sources, highlighting fundamental challenges in seedless extraction.

Findings

Constructed explicit extractors for degree   2 polynomial sources

Established limitations of sumset extractors for low min-entropy polynomial sources

Introduced polynomial NOBF sources as a new algebraic source family

Abstract

We explicitly construct the first nontrivial extractors for degree $d \ge 2$ polynomial sources over $\mathbb{F}_2^n$. Our extractor requires min-entropy $k\geq n - \tilde{\Omega}(\sqrt{\log n})$. Previously, no constructions were known, even for min-entropy $k\geq n-1$. A key ingredient in our construction is an input reduction lemma, which allows us to assume that any polynomial source with min-entropy $k$ can be generated by $O(k)$ uniformly random bits. We also provide strong formal evidence that polynomial sources are unusually challenging to extract from, by showing that even our most powerful general purpose extractors cannot handle polynomial sources with min-entropy below $k\geq n-o(n)$. In more detail, we show that sumset extractors cannot even disperse from degree $2$ polynomial sources with min-entropy $k\geq n-O(n/\log\log n)$. In fact, this impossibility result even holds for a more specialized family of sources that we introduce, called polynomial non-oblivious bit-fixing (NOBF) sources. Polynomial NOBF sources are a natural new family of algebraic sources that lie at the intersection of polynomial and variety sources, and thus our impossibility result applies to both of these classical settings. This is especially surprising, since we do have variety extractors that slightly beat this barrier - implying that sumset extractors are not a panacea in the world of seedless extraction.