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.