Algebraic Pseudorandomness in $VNC^0$

TL;DR

This paper constructs explicit pseudorandom generators computable in constant-depth arithmetic formulas, advancing derandomization techniques for polynomial identity testing and deriving lower bounds for algebraic proof systems.

Contribution

It introduces new $VNC^0$-computable hitting set generators based on circuit complexity lower bounds, not just degree bounds, for algebraic circuit classes.

Findings

Unconditional $VNC^0$-computable generator hits constant-depth polynomial-size circuits.

Conditional generators hit formulas and branching programs under hardness assumptions.

Derives lower bounds for subsystems of the Geometric Ideal Proof System.

Abstract

We study the arithmetic complexity of hitting set generators, which are pseudorandom objects used for derandomization of the polynomial identity testing problem. We give new explicit constructions of hitting set generators whose outputs are computable in $VNC^0$, i.e., can be computed by arithmetic formulas of constant size. Unconditionally, we construct a $VNC^0$-computable generator that hits arithmetic circuits of constant depth and polynomial size. We also give conditional constructions, under strong but plausible hardness assumptions, of $VNC^0$-computable generators that hit arithmetic formulas and arithmetic branching programs of polynomial size, respectively. As a corollary of our constructions, we derive lower bounds for subsystems of the Geometric Ideal Proof System of Grochow and Pitassi. Constructions of such generators are implicit in prior work of Kayal on lower bounds for the degree of annihilating polynomials. Our main contribution is a construction whose correctness relies on circuit complexity lower bounds rather than degree lower bounds.