Exponential Correlation Bounds for Polynomials

2026-09-23

AI Summary

This study addresses the fundamental challenge in computational complexity of establishing correlation bounds between low-degree polynomials and Boolean functions, alongside the efficient construction of pseudorandom generators (PRGs). By integrating polynomial analysis over finite fields, combinatorics, and circuit complexity theory, this work rigorously proves an exponential correlation bound between the XOR of k disjoint block-majority functions and degree-d polynomials over finite fields. Leveraging this bound, the paper overcomes existing limitations on PRG seed lengths by constructing a novel PRG with merely polylogarithmic seed length. This construction achieves efficient pseudorandomization against both low-degree polynomials and constant-depth alternating circuits augmented with parity gates, thereby advancing the state of the art in derandomization and circuit lower bounds.

Abstract

We prove that the XOR of $k$ majorities on disjoint blocks of \(\ell\) bits has correlation at most \((2d/\sqrt{\ell})^k\) with every degree-\(d\) polynomial over \(\mathbb F_2\). By known techniques, this implies pseudorandom generators with polylogarithmic seed length for low-degree polynomials over $\mathbb F_2$ and for alternating circuits with parity gates.

Problem

• correlation bounds
• low-degree polynomials
• pseudorandom generators
• majority functions
• alternating circuits

Innovation

• Exponential Correlation Bounds
• Pseudorandom Generators
• Low-degree Polynomials
• Majority Functions
• Alternating Circuits