Deterministic Extractors for Additive Sources

TL;DR

This paper introduces explicit randomness extractors for additive sources like APs, GAPs, and Bohr sets, generalizing affine sources and improving previous constructions, with applications over finite fields.

Contribution

It provides the first explicit extractors for additive sources with linear min-entropy, extending to various structured sources and improving on prior affine source results.

Findings

Explicit extractors for additive sources with linear min-entropy.

Improved construction of line sources requiring smaller field size.

Generalization of results to APs and GAPs.

Abstract

We propose a new model of a weakly random source that admits randomness extraction. Our model of additive sources includes such natural sources as uniform distributions on arithmetic progressions (APs), generalized arithmetic progressions (GAPs), and Bohr sets, each of which generalizes affine sources. We give an explicit extractor for additive sources with linear min-entropy over both $\mathbb{Z}_p$ and $\mathbb{Z}_p^n$, for large prime $p$, although our results over $\mathbb{Z}_p^n$ require that the source further satisfy a list-decodability condition. As a corollary, we obtain explicit extractors for APs, GAPs, and Bohr sources with linear min-entropy, although again our results over $\mathbb{Z}_p^n$ require the list-decodability condition. We further explore special cases of additive sources. We improve previous constructions of line sources (affine sources of dimension 1), requiring a field of size linear in $n$, rather than $\Omega(n^2)$ by Gabizon and Raz. This beats the non-explicit bound of $\Theta(n \log n)$ obtained by the probabilistic method. We then generalize this result to APs and GAPs.