Algebraic Pseudorandomness in $VNC^0$

2025-05-15

AI Summary

This work addresses the deterministic derandomization of Polynomial Identity Testing (PIT) in algebraic complexity theory, focusing on constructing explicit hitting set generators computable by $VNC^0$—i.e., constant-depth arithmetic formulas. Departing from conventional approaches reliant on degree-based lower bounds, the paper establishes the existence of $VNC^0$-computable hitting sets via circuit complexity lower bounds, yielding the first unconditional explicit construction for constant-depth arithmetic circuits. Under reasonable hardness assumptions, the construction extends to general arithmetic formulas and branching programs. Key contributions include: (1) the first explicit hitting set generator computable in $VNC^0$; (2) the first nontrivial lower bound for a subsystem of the Geometric Ideal Proof System (GIPS); and (3) a new paradigm for algebraic pseudorandomness grounded in circuit complexity lower bounds, unifying hardness-to-randomness tradeoffs in the algebraic setting.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

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.

Problem

• Study arithmetic complexity of hitting set generators
• Construct VNC^0-computable generators for derandomization
• Derive lower bounds for Geometric Ideal Proof System

Innovation

• Constructs VNC^0-computable hitting set generators
• Uses constant-size arithmetic formulas
• Relies on circuit complexity lower bounds