The Correlation of Multiplicative Characters with Polynomials over Finite Fields

This talk will focus on the complexity of the cubic-residue (and higher-residue) characters over GF(2^n) , in the context of both arithmetic circuits and polynomials.

We show that no subexponential-size, constant-depth arithmetic circuit over GF(2) can correctly compute the cubic-residue character for more than 1/3 + o(1) fraction of the elements of GF(2^n). The key ingredient in the proof is an adaptation of the Razborov-Smolensky method for circuit lower bounds to setting of univariate polynomials over GF(2^n) .

As a corollary, we show that the cubic-residue character over GF(2^n) is uncorrelated with all degree-d n-variate GF(2) polynomials (viewed as functions over GF(2^n) in a natural way), provided d << n^{0.1} . Classical approaches based on van der Corput differencing and the Weil bounds show this for d << log(n).