XOR Lemmas for Resilient Functions against Polynomials
Electron. Colloquium Comput. Complex.(2020)
Abstract
A major challenge in complexity theory is to explicitly construct functions that have small correlation with low-degree polynomials over F-2. We introduce a new technique to prove such correlation bounds with F-2 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 F-2. We show that for any n-variate polynomial.. over F-2 of degree at most d, there is a small set S subset of [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.
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Key words
Pseudorandom Generators,Low-degree polynomials,Correlation bounds,Resilient functions
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