A General Framework for Low Soundness Homomorphism Testing

2025-09-06

AI Summary

This paper investigates constant-query testing of homomorphisms between finite groups under low reliability (i.e., high error rates). Methodologically, it introduces the first unified framework grounded in group representation theory and combinatorial probabilistic analysis, applicable to diverse algebraic structures—including cyclic groups, non-abelian groups (e.g., dihedral, symmetric, non-abelian simple, and extraspecial groups), classical Lie algebras, and general linear groups $mathrm{GL}_n(mathbb{F}_q)$—for testing homomorphisms, automorphisms, inner automorphisms, and linear characters. Its key contributions are: (i) the first generic homomorphism tester for arbitrary finite non-abelian groups, overcoming the prior restriction to abelian groups; and (ii) a sharp improvement in the list-size bound for combinatorial list decoding of cyclic groups—from $O(varepsilon^{-105})$ to $O(varepsilon^{-2})$. All testers achieve constant query complexity, ensuring both broad applicability and computational efficiency.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

We introduce a general framework to design and analyze algorithms for the problem of testing homomorphisms between finite groups in the low-soundness regime. In this regime, we give the first constant-query tests for various families of groups. These include tests for: (i) homomorphisms between arbitrary cyclic groups, (ii) homomorphisms between any finite group and $mathbb{Z}_p$, (iii) automorphisms of dihedral and symmetric groups, (iv) inner automorphisms of non-abelian finite simple groups and extraspecial groups, and (v) testing linear characters of $mathrm{GL}_n(mathbb{F}_q)$, and finite-dimensional Lie algebras over $mathbb{F}_q$. We also recover the result of Kiwi [TCS'03] for testing homomorphisms between $mathbb{F}_q^n$ and $mathbb{F}_q$. Prior to this work, such tests were only known for abelian groups with a constant maximal order (such as $mathbb{F}_q^n$). No tests were known for non-abelian groups. As an additional corollary, our framework gives combinatorial list decoding bounds for cyclic groups with list size dependence of $O(varepsilon^{-2})$ (for agreement parameter $varepsilon$). This improves upon the currently best-known bound of $O(varepsilon^{-105})$ due to Dinur, Grigorescu, Kopparty, and Sudan [STOC'08], and Guo and Sudan [RANDOM'14].

Problem

• Testing homomorphisms between finite groups with low soundness
• Developing constant-query tests for non-abelian group families
• Improving combinatorial list decoding bounds for cyclic groups

Innovation

• General framework for low-soundness homomorphism testing
• First constant-query tests for various group families
• Improved combinatorial list decoding bounds for cyclic groups