Algebraic Hardness versus Randomness in Low Characteristic

TL;DR

This paper demonstrates that lower bounds for explicit polynomials over fields of positive characteristic can be used to derandomize polynomial identity testing without restrictions on the characteristic, using standard hardness notions.

Contribution

It introduces a method combining the Kabanets-Impagliazzo generator with a white-box p-th root extraction to achieve derandomization in positive characteristic fields, extending previous results.

Findings

Derandomization of polynomial identity testing over fields of positive characteristic.

Efficient p-th root extraction for circuits computing p-th powers when variables are bounded.

Near-complete derandomization results using explicit constant-variate polynomials.

Abstract

We show that lower bounds for explicit constant-variate polynomials over fields of characteristic $p > 0$ are sufficient to derandomize polynomial identity testing over fields of characteristic $p$. In this setting, existing work on hardness-randomness tradeoffs for polynomial identity testing requires either the characteristic to be sufficiently large or the notion of hardness to be stronger than the standard syntactic notion of hardness used in algebraic complexity. Our results make no restriction on the characteristic of the field and use standard notions of hardness. We do this by combining the Kabanets-Impagliazzo generator with a white-box procedure to take $p$-th roots of circuits computing a $p$-th power over fields of characteristic $p$. When the number of variables appearing in the circuit is bounded by some constant, this procedure turns out to be efficient, which allows us to bypass difficulties related to factoring circuits in characteristic $p$. We also combine the Kabanets-Impagliazzo generator with recent "bootstrapping" results in polynomial identity testing to show that a sufficiently-hard family of explicit constant-variate polynomials yields a near-complete derandomization of polynomial identity testing. This result holds over fields of both zero and positive characteristic and complements a recent work of Guo, Kumar, Saptharishi, and Solomon, who obtained a slightly stronger statement over fields of characteristic zero.