Low-Degree Polynomials Are Good Extractors

2024-05-16

Electron. Colloquium Comput. Complex.

AI Summary

This paper addresses the universality of random low-degree polynomials over 𝔽₂ as strong extractors. Prior work established extractor guarantees only for uniform, affine, or local sources; this work is the first to unify treatment of polynomial sources, algebraic varieties (variety sources), and sumset sources. We introduce a novel construction based on truncated Reed–Muller codes and a new extractor reduction technique. Our theoretical contributions are: (1) an ε-strong extractor for n-bit sumset sources with min-entropy k = O((n/ε²)²⁄ᵈ), breaking previous source-class restrictions; (2) improved extraction bounds for local sources; and (3) nearly tight error bounds, a new separation for linear read-once branching programs (ROBPs), and a strengthened impossibility theorem for sumset extractors. The analysis integrates tools from finite-field polynomial theory, coding theory, and min-entropy estimation.

Technology Category

• Machine Learning: Other Foundations of Machine Learning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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).

Problem

• Proving low-degree polynomials extract randomness from diverse distributions
• Generalizing extractor results beyond uniform, affine, and local sources
• Establishing tight bounds for sumset sources with low-degree polynomials

Innovation

• Random low-degree polynomials extract randomness broadly
• Unified approach for small families of sources
• Novel reduction and structural analysis techniques