Automorphisms of Finite Rings and Applications to Complexity of Problems

In mathematics, automorphisms of algebraic structures play an important role. Automorphisms capture the symmetries inherent in the structures and many important results have been proved by analyzing the automorphism group of the structure. For example, Galois characterized degree five univariate polynomials f over rationals whose roots can be expressed using radicals (using addition, subtraction, multiplication, division and taking roots) via the structure of automorphism group of the splitting field of f . In computer science too, automorphisms have played a useful role in our understanding of the complexity of many algebraic problems. From a computer science perspective, perhaps the most important structure is that of finite rings. This is because a number of algebraic problems efficiently reduce to questions about automorphisms and isomorphisms of finite rings. In this paper, we collect several examples of this from the literature as well as providing some new and interesting connections. As discussed in section 2, finite rings can be represented in several ways. We will be primarily interested in the basis representation where the ring is specified by its basis under addition. For this representation, the complexity of deciding most of the questions about the automorphisms and isomorphisms is in FP [KS04]. For example, finding ring automorphism (find a non-trivial automorphism of a ring), automorphism counting problem (count the number of automorphisms of a ring), ring isomorphism problem (decide if two rings are isomorphic), finding ring isomorphism (find an isomorphism between two rings). Also, ring automorphism problem (decide if a ring has a non-trivial automorphism) is in P [KS04]. In addition, a number of problems can be reduced to answering these questions. Some of them are: