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

TL;DR

This paper explores the computational complexity of finding the unit group in finite rings, establishing equivalences with key number theory problems like factoring and discrete logarithms.

Contribution

It demonstrates that computing the unit group and related algebraic structures in finite rings is computationally equivalent to fundamental number theory problems.

Findings

Unit group computation is equivalent to integer factoring.

Determining the abelianization of the unit group relates to discrete logarithms.

The results connect algebraic problems in finite rings with classical number theory challenges.

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.