Optimal PRGs for Low-Degree Polynomials over Polynomial-Size Fields

2026-02-10

AI Summary

This work addresses the challenge of constructing seed-length-optimal pseudorandom generators (PRGs) for low-degree polynomials over small finite fields. Building upon the Derksen–Viola framework, the authors introduce a novel pseudorandom object that replaces traditional hitting set generators, substantially relaxing the requirement on field size. Under the assumption of sufficiently large characteristic, they achieve—for the first time—a PRG with optimal seed length for degree-$d$ polynomials over fields of size $q \approx d^4$. Moreover, the paper uncovers a threshold phenomenon linking field size to PRGs over the binary field and demonstrates that this threshold is inherent: if a PRG without characteristic restrictions could be constructed over sublinear field sizes (e.g., $q = d^{0.99}$), it would directly yield a near-optimal PRG over the binary field.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• pseudorandom generators
• low-degree polynomials
• finite fields
• seed length
• derandomization

Innovation

• pseudorandom generators
• low-degree polynomials
• finite fields
• optimal seed length
• hitting-set generator