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Low-Degree Polynomials Are Good Extractors

Omar Alrabiah, Jesse Goodman, Jonathan Mosheiff, João Ribeiro

Abstract

We prove that random low-degree polynomials (over $\mathbb{F}_2$) are unbiased, in an extremely general sense. That is, we show that random low-degree polynomials are good randomness extractors for a wide class of distributions. Prior to our work, such results were only known for the small families of (1) uniform sources, (2) affine sources, and (3) local sources. We significantly generalize these results, and prove the following. 1. Low-degree polynomials extract from small families. We show that a random low-degree polynomial is a good low-error extractor for any small family of sources. In particular, we improve the positive result of Alrabiah, Chattopadhyay, Goodman, Li, and Ribeiro (ICALP 2022) for local sources, and give new results for polynomial and variety sources via a single unified approach. 2. Low-degree polynomials extract from sumset sources. We show that a random low-degree polynomial is a good extractor for sumset sources, which are the most general large family of sources (capturing independent sources, interleaved sources, small-space sources, and more). Formally, for any even $d$, we show that a random degree $d$ polynomial is an $\varepsilon$-error extractor for $n$-bit sumset sources with min-entropy $k=O(d(n/\varepsilon^2)^{2/d})$. This is nearly tight in the polynomial error regime. Our results on sumset extractors imply new complexity separations for linear ROBPs, and the tools that go into its proof may be of independent interest. The two main tools we use are a new structural result on sumset-punctured Reed-Muller codes, paired with a novel type of reduction between extractors. Using the new structural result, we obtain new limits on the power of sumset extractors, strengthening and generalizing the impossibility results of Chattopadhyay, Goodman, and Gurumukhani (ITCS 2024).

Low-Degree Polynomials Are Good Extractors

Abstract

We prove that random low-degree polynomials (over ) are unbiased, in an extremely general sense. That is, we show that random low-degree polynomials are good randomness extractors for a wide class of distributions. Prior to our work, such results were only known for the small families of (1) uniform sources, (2) affine sources, and (3) local sources. We significantly generalize these results, and prove the following. 1. Low-degree polynomials extract from small families. We show that a random low-degree polynomial is a good low-error extractor for any small family of sources. In particular, we improve the positive result of Alrabiah, Chattopadhyay, Goodman, Li, and Ribeiro (ICALP 2022) for local sources, and give new results for polynomial and variety sources via a single unified approach. 2. Low-degree polynomials extract from sumset sources. We show that a random low-degree polynomial is a good extractor for sumset sources, which are the most general large family of sources (capturing independent sources, interleaved sources, small-space sources, and more). Formally, for any even , we show that a random degree polynomial is an -error extractor for -bit sumset sources with min-entropy . This is nearly tight in the polynomial error regime. Our results on sumset extractors imply new complexity separations for linear ROBPs, and the tools that go into its proof may be of independent interest. The two main tools we use are a new structural result on sumset-punctured Reed-Muller codes, paired with a novel type of reduction between extractors. Using the new structural result, we obtain new limits on the power of sumset extractors, strengthening and generalizing the impossibility results of Chattopadhyay, Goodman, and Gurumukhani (ITCS 2024).
Paper Structure (35 sections, 46 theorems, 114 equations)

This paper contains 35 sections, 46 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Pseudorandomness
  3. Coding theory
  4. Complexity theory
  5. Our results
  6. Low-degree polynomials extract from small families
  7. Concurrent work
  8. Low-degree polynomials extract from sumset sources
  9. Linear ROBPs
  10. Sumset-punctured Reed-Muller codes
  11. A novel reduction between extractors
  12. Overview of our techniques
  13. Low-degree polynomials extract from small families
  14. Low-degree polynomials extract from sumset sources
  15. Low-degree polynomials disperse from sumset sources
  16. ...and 20 more sections

Key Result

Theorem 0

For every $\delta\in(0,1)$ there is a constant $c>0$ such that the following holds. Let $d\in\mathbb{N}$ be an integer satisfying $1\leq d\leq(1-\delta)k$. Then for a random degree $d$ polynomial $f:\mathbb{F}_2^n\to \mathbb{F}_2$,

Theorems & Definitions (91)

  • Theorem 0: Random low-degree polynomials are unbiased BHL12
  • Definition 1: Randomness extractor
  • Theorem 1: Low-degree polynomials extract from a single source
  • Corollary 1: Low-degree polynomials extract from small families
  • Theorem 2: Low-degree polynomials extract from sumset sources
  • Claim 1: Informal
  • Definition 2: Statistical distance
  • Definition 3: Min-entropy
  • Definition 4: $(n,k)$-source
  • Definition 5: $(n,k)$-sumset source
  • ...and 81 more