Correlation Bounds for Polynomials

Construct an explicit function f: F2^n → F2, computable in NP, such that for every F2-polynomial p: F2^n → F2 of degree at most log2 n, the correlation satisfies |E[(-1)^{f(x)}(-1)^{p(x)}]| ≤ 1/n.

Background

This problem asks for an explicit Boolean function that has very low correlation with every low-degree polynomial over F2, specifically for degree up to log2 n. Such functions are central to circuit lower bounds and pseudorandomness against low-degree polynomials.

Smolensky showed that the mod3 function has exponentially small correlation against degree up to c√n, but known techniques face barriers below 1/√n. There are also functions (in P) with exponentially small correlation against degree up to about 0.99 log n, but finding an NP-explicit function with correlation ≤ 1/n against degree up to log n remains unresolved.