Involutory Matrices Over Finite Local Rings

Abstract

A square matrix A over a commutative ring R is said to be involutory if A2 = I (identity matrix). It has been recognized for some time that involutory matrices have important applications in algebraic cryptography and the special cases where R is either a finite field or a quotient ring of the rational integers have been extensively researched. However, there has been no detailed attempt to extend the theory to all finite commutative rings. In this paper we illustrate in detail the theory of involutory matrices over finite commutative rings with 1 having odd characteristic. The method is a careful analysis of finite local rings of odd prime power characteristic. The techniques might be also used in the examination of involutory matrices over local rings of characteristic 2λ; however, as illustrated by finite fields of characteristic 2 and Z/2λZ (Z the rational integers), the arguments are basically different. The reader will note the methods are not limited to only questions on involutory matrices.