Worst Case to Average Case Reductions for Polynomials

TL;DR

This paper proves that bias implies low rank for polynomials over all fields, enabling worst-case to average-case reductions and impacting pseudorandom generator construction and concise representation testing.

Contribution

It confirms Green and Tao's conjecture that bias implies low rank over all fields, leading to new worst-case to average-case reduction techniques for polynomials.

Findings

Bias implies low rank over all fields.

Polynomial approximation can be converted into polynomial computation.

Implications for pseudorandom generators and concise representation testing.

Abstract

A degree-$d$ polynomial $p$ in $n$ variables over a field $\F$ is {\em equidistributed} if it takes on each of its $|\F|$ values close to equally often, and {\em biased} otherwise. We say that $p$ has a {\em low rank} if it can be expressed as a bounded combination of polynomials of lower degree. Green and Tao [gt07] have shown that bias imply low rank over large fields (i.e. for the case $d < |\F|$). They have also conjectured that bias imply low rank over general fields. In this work we affirmatively answer their conjecture. Using this result we obtain a general worst case to average case reductions for polynomials. That is, we show that a polynomial that can be {\em approximated} by few polynomials of bounded degree, can be {\em computed} by few polynomials of bounded degree. We derive some relations between our results to the construction of pseudorandom generators, and to the question of testing concise representations.