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.

翻译：本文研究有限环单位群的计算确定问题，即计算其有限表示并给出将单位元表示为生成元中字的算法。我们证明该问题等价于整数分解与有限域上离散对数求解这两类数论问题。对于确定单位群的阿贝尔化或有限环的第一 $K$-群问题，我们同样证明了类似的等价性。