Low-Degree Polynomials Extract from Local Sources

by   Omar Alrabiah, et al.

We continue a line of work on extracting random bits from weak sources that are generated by simple processes. We focus on the model of locally samplable sources, where each bit in the source depends on a small number of (hidden) uniformly random input bits. Also known as local sources, this model was introduced by De and Watson (TOCT 2012) and Viola (SICOMP 2014), and is closely related to sources generated by 𝖠𝖢^0 circuits and bounded-width branching programs. In particular, extractors for local sources also work for sources generated by these classical computational models. Despite being introduced a decade ago, little progress has been made on improving the entropy requirement for extracting from local sources. The current best explicit extractors require entropy n^1/2, and follow via a reduction to affine extractors. To start, we prove a barrier showing that one cannot hope to improve this entropy requirement via a black-box reduction of this form. In particular, new techniques are needed. In our main result, we seek to answer whether low-degree polynomials (over 𝔽_2) hold potential for breaking this barrier. We answer this question in the positive, and fully characterize the power of low-degree polynomials as extractors for local sources. More precisely, we show that a random degree r polynomial is a low-error extractor for n-bit local sources with min-entropy Ω(r(nlog n)^1/r), and we show that this is tight. Our result leverages several new ingredients, which may be of independent interest. Our existential result relies on a new reduction from local sources to a more structured family, known as local non-oblivious bit-fixing sources. To show its tightness, we prove a "local version" of a structural result by Cohen and Tal (RANDOM 2015), which relies on a new "low-weight" Chevalley-Warning theorem.


page 1

page 2

page 3

page 4

∙ 09/20/2023

Extractors for Polynomial Sources over 𝔽_2

We explicitly construct the first nontrivial extractors for degree d ≥ 2...
∙ 10/25/2021

Extractors for Sum of Two Sources

We consider the problem of extracting randomness from sumset sources, a ...
∙ 07/15/2020

Explicit Designs and Extractors

We give significantly improved explicit constructions of three related p...
∙ 05/28/2019

Average Bias and Polynomial Sources

We identify a new notion of pseudorandomness for randomness sources, whi...
∙ 11/26/2022

Extractors for Images of Varieties

We construct explicit deterministic extractors for polynomial images of ...
∙ 06/29/2023

Extracting Mergers and Projections of Partitions

We study the problem of extracting randomness from somewhere-random sour...
∙ 09/18/2017

Local decoding and testing of polynomials over grids

The well-known DeMillo-Lipton-Schwartz-Zippel lemma says that n-variate ...

Please sign up or login with your details

Forgot password? Click here to reset