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TR10-033 | 6th March 2010 15:04
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#### Pseudorandom generators for $\mathrm{CC}_0[p]$ and the Fourier spectrum of low-degree polynomials over finite fields

**Abstract:**
In this paper we give the first construction of a pseudorandom generator, with seed length $O(\log n)$, for $\mathrm{CC}_0[p]$, the class of constant-depth circuits with unbounded fan-in $\mathrm{MOD}_p$ 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 [LVW93] or that could only fool the distribution generated by linear functions over $\mathbb{F}_p$, when evaluated on the Boolean cube [LRTV09,MZ09].

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 $\mathbb{F}_p$.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, up to error $\delta$, by a function of a small number (depending only on $\epsilon,\delta$ and $d$) of lower degree polynomials.