XOR Lemmas for Resilient Functions Against Polynomials.

A major challenge in complexity theory is to explicitly construct functions that have small correlation with low-degree polynomials over F2. We introduce a new technique to prove such correlation bounds with F2 polynomials. Using this technique, we bound the correlation of an XOR of Majorities with constant degree polynomials. In fact, we prove a more general XOR lemma that extends to arbitrary resilient functions. We conjecture that the technique generalizes to higher degree polynomials as well. A key ingredient in our new approach is a structural result about the Fourier spectrum of low degree polynomials over F2. We show that for any n-variate polynomial p over F2 of degree at most d, there is a small set S ⊂ [n] of variables, such that almost all of the Fourier mass of p lies on Fourier coefficients that intersect with S. In fact our result is more general, and finds such a set S for any low-dimensional subspace of polynomials. This generality is crucial in deriving the new XOR lemmas.