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 MODp gates, for some prime p. More accurately, the seed length of our generator is O(log n) for any constant error ∈ > 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 (Luby et al. 1993) or could only fool the distribution generated by linear functions over F_p, when evaluated on the Boolean cube (Lovett et al. 2009; Meka & Zuckerman 2009). En route 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 Fp. Then, for every ∈ > 0, there exists a subset S ⊂ [n], whose size depends only on d and ∈, such that (Formula presented.). 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 Fp. If the distribution of f when applied to uniform zero-one bits is ∈-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 d, by a function of a small number (depending only on ∈, δ and δ) of lower degree polynomials.