Pseudorandom generators for CC⁰[p] and the fourier spectrum of low-degree polynomials over finite fields

In this paper we give the first construction of a pseudorandom generator, with seed length O(log n), for CC⁰[p], the class of constant-depth circuits with unbounded fan-in MOD_p gates, for some prime p. More accurately, the seed length of our generator is O(log n) for any constant error ε > 0. In fact, we obtain our generator by fooling distributions generated by low degree polynomials, over double-struck F_p, when evaluated on the Boolean cube. This result significantly extends previous constructions that either required a long seed [1] or that could only fool the distribution generated by linear functions over double-struck F_p, when evaluated on the Boolean cube [2], [3]. Enroute of constructing our PRG, we prove two structural results for low degree polynomials over finite fields that can be of independent interest. 1) Let f be an n-variate degree d polynomial over double-struck F_p. Then, for every ε > 0 there exists a subset S ⊂ [n], whose size depends only on d and ε, such that ∑_{α∈double-struck Fpn:α≠0,S=0}|f(α)| ² ≤ ε. Namely, there is a constant size subset S such that the total weight of the nonzero Fourier coefficients that do not involve any variable from S is small. 2) Let f be an n-variate degree d polynomial over double-struck F_p. If the distribution of f when applied to uniform zero-one bits is ε-far (in statistical distance) from its distribution when applied to biased bits, then for every δ > 0, f can be approximated over zero-one bits, up to error δ, by a function of a small number (depending only on ε, δ and d) of lower degree polynomials.