, Volume 22, Issue 4, pp 679-725
Date: 23 Nov 2012

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

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Abstract

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 \({\epsilon > 0}\) . In fact, we obtain our generator by fooling distributions generated by low-degree polynomials, over \({\mathbb{F}_p}\) , 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 \({\mathbb{F}_p}\) , 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 \({\mathbb{F}_p}\) . Then, for every \({\epsilon > 0}\) , there exists a subset \({S \subset [n]}\) , whose size depends only on d and \({\epsilon}\) , such that \({\sum_{\alpha \in \mathbb{F}_p^n: \alpha \ne 0, \alpha_S=0}|\hat{f}(\alpha)|^2 \leq \epsilon}\) . 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 \({\mathbb{F}_p}\) . If the distribution of f when applied to uniform zero–one bits is \({\epsilon}\) -far (in statistical distance) from its distribution when applied to biased bits, then for every \({\delta > 0}\) , f can be approximated over zero–one bits, up to error δ, by a function of a small number (depending only on \({\epsilon,\delta}\) and d) of lower degree polynomials.