Abstract

We give an explicit construction of pseudorandom generators against low degree polynomials over finite fields. We show that the sum of 2d small-biased generators with error e2O(d) is a pseudorandom generator against degree d polynomials with error e. This gives a generator with seed length 2O(d) log(n/e). Our construction follows the recent breakthrough result of Bogadnov and Viola. Their work shows that the sum of d small-biased generators is a pseudo-random generator against degree d polynomials, assuming the Inverse Gowers Conjecture. However, this conjecture is only proven for d=2,3. The main advantage of our work is that it does not rely on any unproven conjectures.

Highlights

  • We are interested in explicitly constructing pseudorandom generators (PRG) against low-degree polynomials over small finite fields

  • We are interested in PRGs that are pseudorandom against all degree-d polynomials with error ε, and use as few random bits as possible

  • We prove that it is enough to be pseudorandom against ∆p in order to be pseudorandom against p(x + y), and that it is sufficient to have x, x, x, y, y and y come from a PRG that is pseudorandom against degree-(d − 1) polynomials

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Summary

Introduction

We are interested in explicitly constructing pseudorandom generators (PRG) against low-degree polynomials over small finite fields. The case of pseudorandom generators against linear polynomials, usually called small-bias generators (or epsilon-biased generators, a term we do not use in this paper to avoid confusion), was first studied (over F = F2) by Naor and Naor [14] and later by Alon, Goldreich, Hastad and Peralta [1] They and others gave explicit constructions, which were later generalized to arbitrary finite fields. Bogdanov and Viola [7] presented a novel approach for constructing a PRG for low-degree polynomials over small fields Their construction is the sum of d independent small-bias generators. They showed that, if a conjecture in additive combinatorics called the inverse conjecture for the Gowers norm holds, their construction is a PRG for degree-d polynomials. UNCONDITIONAL PSEUDORANDOM GENERATORS FOR LOW-DEGREE POLYNOMIALS with error ε using 2cd log(| F |n/ε) random bits for the seed

Overview of proof method
Subsequent work
Preliminaries
Main theorem
Case I
Case II
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