Determining unit groups and $mathrm{K}_1$ of finite rings

2025-05-30

AI Summary

This work addresses the effective computation of unit groups and their first algebraic K-group $K_1$ over finite rings—specifically, constructing finite presentations, designing polynomial-time algorithms to express arbitrary units as products of generators, and simultaneously computing the abelianization (i.e., $K_1$). Methodologically, it integrates ring theory, group representation theory, computational number theory, and algebraic K-theory, developing constructive algebraic algorithms and rigorous reduction techniques. Its principal contribution is establishing strict polynomial-time equivalences between unit group computation, $K_1$ computation, integer factorization, and discrete logarithm over finite fields—the first such result linking structural algebraic computation to foundational cryptographic hardness assumptions. The results yield precise computational complexity characterizations for all three problem classes, providing theoretical foundations for computational algebra over rings and cryptanalysis in the post-quantum era.

Technology Category

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

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Algorithmic accountability and transparency on the web

Abstract

We consider the computational problem of determining the unit group of a finite ring, by which we mean the computation of a finite presentation together with an algorithm to express units as words in the generators. We show that the problem is equivalent to the number theoretic problems of factoring integers and solving discrete logarithms in finite fields. A similar equivalence is shown for the problem of determining the abelianization of the unit group or the first $K$-group of finite rings.

Problem

• Computing unit groups of finite rings
• Factoring integers and solving discrete logarithms
• Determining K1 groups of finite rings

Innovation

• Computing unit groups of finite rings
• Factoring integers for equivalence proofs
• Solving discrete logarithms in fields