Extractors for Polynomial Sources over $\mathbb{F}_2$
Eshan Chattopadhyay, Jesse Goodman, Mohit Gurumukhani
TL;DR
This work initiates seedless extraction for polynomial sources over $\mathbb{F}_2$ by constructing the first nontrivial explicit extractors for degree $d\ge 2$ and near-full min-entropy, using a novel input-reduction lemma that reduces an arbitrary source to a convex combination of short-input instances. The core approach combines existential random-function arguments with a derandomization via $t$-wise independent hash families, yielding a poly-time explicit extractor with output $\Omega(\log\log n)$ bits under min-entropy $k \ge n - \Omega\big(\frac{\sqrt{\log n}}{(\log \log n / d)^{d/2}}\big)$. The paper also proves strong impossibility results showing that sumset extractors cannot disperse from quadratic polynomial NOBF sources with $k \ge n - O(n/\log\log n)$, highlighting intrinsic difficulties in polynomial/algebraic sources and the limits of seedless extraction for these classes. Together, these results advance the theory of algebraic sources and have implications for extracting from circuit-like models and variety-based sources, clarifying both achievable directions and inherent barriers.
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 $k\geq n - \tildeΩ(\sqrt{\log n})$. 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.