Optimal PRGs for Low-Degree Polynomials over Polynomial-Size Fields
Gil Cohen, Dean Doron, Noam Goldgraber
TL;DR
The paper advances pseudorandomness for low-degree polynomials by constructing the first explicit PRG with optimal seed length for degree $d$-polynomials over fields of polynomial size, achieving seed length $s = O(d\log n + \log q)$ when $q = \Omega((d\log d)^4/\varepsilon^2)$ and ${\rm char}({\mathbb F}_q) = \Omega(d^2)$. The key methodological shift is replacing hitting-set generators with polynomial hitting-set generators (PHSGs) within the Derksen–Viola restriction-map framework, enabling high-density restrictions at field sizes polynomial in $d$ and removing dependence on $n$ in the field-size requirement. This establishes a threshold phenomenon: improving $q$ from quartic to sublinear in $d$ would, via a reduction, yield a comparable PRG for the binary field, highlighting an inherent barrier in this regime. The construction relies on indecomposability-preserving restriction maps, Lecerf’s technique, Gauss’s lemma, and a two-tier PRG definition combining a PHSG and an HSG; it also presents a pathway to smaller fields via trace-based field reductions, contingent on unproven subtasks about base-field PRGs. Overall, the work sharpens the frontier between small-field and large-field PRGs for low-degree polynomials and offers a concrete and scalable route to optimal seed-length generators over fields of polynomial size.
Abstract
Pseudorandom generators (PRGs) for low-degree polynomials are a central object in pseudorandomness, with applications to circuit lower bounds and derandomization. Viola's celebrated construction gives a PRG over the binary field, but with seed length exponential in the degree $d$. This exponential dependence can be avoided over sufficiently large fields. In particular, Dwivedi, Guo, and Volk constructed PRGs with optimal seed length over fields of size exponential in $d$. The latter builds on the framework of Derksen and Viola, who obtained optimal-seed constructions over fields of size polynomial in $d$, although growing with the number of variables $n$. In this work, we construct the first PRG with optimal seed length for degree-$d$ polynomials over fields of polynomial size, specifically $q \approx d^4$, assuming sufficiently large characteristic. Our construction follows the framework of prior work and reduces the required field size by replacing the hitting-set generator used in previous constructions with a new pseudorandom object. We also observe a threshold phenomenon in the field-size dependence. Specifically, we prove that constructing PRGs over fields of sublinear size, for example $q = d^{0.99}$ where $q$ is a power of two, would already yield PRGs for the binary field with comparable seed length via our reduction, provided that the construction imposes no restriction on the characteristic. While a breakdown of existing techniques has been noted before, we prove that this phenomenon is inherent to the problem itself, irrespective of the technique used.