Recursive MDS matrices over finite commutative rings

Abstract

Recursive MDS matrices are used for the design of linear diffusion layers in lightweight cryptographic applications. Most of the works on the construction of recursive MDS matrices either consider matrices over finite fields or block matrices over GL(m,F2). In the first case, there have been works on the direct construction of recursive MDS matrices. The latter case is hard to deal with because of its non-commutative nature. There has not been any serious attempt to look for recursive MDS matrices over finite commutative rings, in particular over local rings of even characteristic. In this work, we present several methods for the construction of recursive MDS companion matrices over finite commutative rings. The main tools are the simple expressions for the determinant of (generalized) Vandermonde and linearized matrices. We show that the determinant of a linearized matrix over a finite commutative ring of prime characteristic can be expressed in a simple form. We discuss a technique called subring construction with which MDS matrices over product rings can be constructed using MDS matrices over subrings. We give a few examples of recursive MDS companion matrices over local rings of even characteristic. We also discuss some results on the nonexistence of recursive MDS matrices over certain rings for some parameter choices.