## Abstract

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 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.

Original language | English |
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Title of host publication | Proceedings - 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, FOCS 2010 |

Publisher | IEEE Computer Society |

Pages | 695-704 |

Number of pages | 10 |

ISBN (Print) | 9780769542447 |

DOIs | |

State | Published - 2010 |

Externally published | Yes |

Event | 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, FOCS 2010 - Las Vegas, NV, United States Duration: 23 Oct 2010 → 26 Oct 2010 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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ISSN (Print) | 0272-5428 |

### Conference

Conference | 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, FOCS 2010 |
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Country/Territory | United States |

City | Las Vegas, NV |

Period | 23/10/10 → 26/10/10 |

## Keywords

- Constant depth circuits
- Fourier spectrum
- Low degree polynomials
- Pseudorandom generator

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