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 CC
⁰
[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 F
_p
, when evaluated on the Boolean cube. This result significantly extends previous constructions that either required a long seed or that could only fool the distribution generated by linear functions over F
_p
, when evaluated on the Boolean cube. 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 F
_p
. Then, for every ϵ > 0 there exists a subset S ⊂ [n], whose size depends only on d and ϵ, such that Σ
_α∈F
p
ⁿ
_{:α≠0,α
S
=0}
|f̂(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 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.