Parallel Algorithms for Group Isomorphism via Code Equivalence

2026-04-15

AI Summary

This work addresses the isomorphism testing problem for coprime extension groups and central-radical groups, aiming to achieve efficient parallel algorithms. By introducing, for the first time, parallelization techniques from linear code equivalence into group isomorphism testing, and integrating Luks’s group-theoretic framework with the AC circuit model, the authors exploit structural properties of group multiplication tables for optimization. The main contribution lies in achieving isomorphism testing for these two classes of groups within the complexity class AC³. Furthermore, the circuit depth for isomorphism testing of arbitrary central-radical groups is reduced to O(log³n) with circuit size n^{O(log log n)}, significantly improving upon prior results.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Applications of cryptography
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

In this paper, we exhibit $\textsf{AC}^{3}$ isomorphism tests for coprime extensions $H \ltimes N$ where $H$ is elementary Abelian and $N$ is Abelian; and groups where $\text{Rad}(G) = Z(G)$ is elementary Abelian and $G = \text{Soc}^{*}(G)$. The fact that isomorphism testing for these families is in $\textsf{P}$ was established respectively by Qiao, Sarma, and Tang (STACS 2011), and Grochow and Qiao (CCC 2014, SIAM J. Comput. 2017). The polynomial-time isomorphism tests for both of these families crucially leveraged small (size $O(\log |G|)$) instances of Linear Code Equivalence (Babai, SODA 2011). Here, we combine Luks' group-theoretic method for Graph Isomorphism (FOCS 1980, J. Comput. Syst. Sci. 1982) with the fact that $G$ is given by its multiplication table, to implement the corresponding instances of Linear Code Equivalence in $\textsf{AC}^{3}$. As a byproduct of our work, we show that isomorphism testing of arbitrary central-radical groups is decidable using $\textsf{AC}$ circuits of depth $O(\log^3 n)$ and size $n^{O(\log \log n)}$. This improves upon the previous bound of $n^{O(\log \log n)}$-time due to Grochow and Qiao (ibid.).

Problem

• Group Isomorphism
• Code Equivalence
• Parallel Algorithms
• Central-Radical Groups
• AC Complexity

Innovation

• Parallel Algorithms
• Group Isomorphism
• Code Equivalence
• AC^3 Complexity
• Central-Radical Groups