## Abstract

The problem of constructing pseudorandom generators for polynomials of low degree is fundamental in complexity theory and has numerous well-known applications. We study the following question, which is a relaxation of this problem: Is it easier to construct pseudorandom generators, or even hitting-set generators, for polynomials p: F^{n}→ F of degree d if we are guaranteed that the polynomial vanishes on at most an ε> 0 fraction of its inputs? We will specifically be interested in tiny values of ε≪ d/ | F|. This question was first considered by Goldreich and Wigderson (STOC 2014), who studied a specific setting geared for a particular application, and another specific setting was later studied by the third author (CCC 2017). In this work, our main interest is a systematic study of the relaxed problem,in its general form, and we prove results that significantly improve and extend the two previously known results. Our contributions are of two types: ∘ Over fields of size 2 ≤ | F| ≤ poly (n) we show that the seed length of any hitting-set generator for polynomials of degree d≤ n^{. 49} that vanish on at most ε= | F| ^{-}^{t} of their inputs is at least Ω ((d/ t) · log (n)). ∘ Over F_{2}, we show that there exists a (non-explicit) hitting-set generator for polynomials of degree d≤ n^{. 99} that vanish on at most ε= | F| ^{-}^{t} of their inputs with seed length O((d- t) · log (n)). We also show a polynomial-time computable hitting-set generator with seed length O((d- t) · (2 ^{d}^{-}^{t}+ log (n))). In addition, we prove that the problem we study is closely related to the following question: “Does there exist a small set S⊆ F^{n} whose degree-d closure is very large?”, where the degree-d closure of S is the variety induced by the set of degree-d polynomials that vanish on S.

Original language | English |
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Article number | 16 |

Journal | Computational Complexity |

Volume | 31 |

Issue number | 2 |

DOIs | |

State | Published - Dec 2022 |

## Keywords

- 11T06 Polynomials over finite fields
- 68Q87 Probability in computer science (algorithm analysis, random structures, phase transitions, etc.)
- Bounded-Degree Closure
- Hitting-Set Generators
- Polynomials
- Pseudorandom Generators
- Quantified Derandomization