Optimal Pseudorandom Generators for Low-Degree Polynomials Over Moderately Large Fields
Ashish Dwivedi, Zeyu Guo, Ben Lee Volk
TL;DR
The paper addresses constructing explicit pseudorandom generators that fool $n$-variate polynomials of degree at most $d$ over a finite field $\,\mathbb{F}_q$. It combines Bogdanov’s plane-restriction paradigm with Lecerf’s multivariate factoring and Derksen–Viola’s indecomposability insights to achieve an explicit PRG with seed length $O(d\log n+\log q)$ under large enough field size and suitable characteristic, improving prior seed-length bounds and removing field-size dependence on $n$. The construction derandomizes Hypothesis (H) via a degree-$O(d)$ HSG and leverages indecomposability to obtain equidistribution, enabling the final PRG guarantee with controlled error $\varepsilon$ and seed length that is near-optimal in many regimes. A prime-degree variant yields similar seed-length guarantees with a milder field-size requirement, and the work outlines several open directions, including tighter field-size bounds and extensions to broader algebraic test classes. Overall, the result advances explicit PRG design for algebraic tests and has implications for derandomizing low-depth circuits and related algebraic computations.
Abstract
We construct explicit pseudorandom generators that fool $n$-variate polynomials of degree at most $d$ over a finite field $\mathbb{F}_q$. The seed length of our generators is $O(d \log n + \log q)$, over fields of size exponential in $d$ and characteristic at least $d(d-1)+1$. Previous constructions such as Bogdanov's (STOC 2005) and Derksen and Viola's (FOCS 2022) had either suboptimal seed length or required the field size to depend on $n$. Our approach follows Bogdanov's paradigm while incorporating techniques from Lecerf's factorization algorithm (J. Symb. Comput. 2007) and insights from the construction of Derksen and Viola regarding the role of indecomposability of polynomials.