On the construction of Cauchy MDS matrices over Galois rings via nilpotent elements and Frobenius maps

TL;DR

This paper presents a novel method for constructing Cauchy MDS matrices over Galois rings using nilpotent elements, Frobenius maps, and automorphisms, reducing matrix entries and generating numerous MDS-preserving functions.

Contribution

It introduces a new approach leveraging nilpotent elements and Frobenius automorphisms to construct and generate Cauchy MDS matrices over Galois rings.

Findings

Constructed a large family of MDS matrices over Galois rings.

Reduced the number of entries in Cauchy MDS matrices.

Generated numerous MDS-preserving functions using Frobenius automorphisms.

Abstract

Let $s,m$ be the positive integers and $p$ be any prime number. Next, let $GR(p^s,p^{sm})$ be a Galois ring of characteristic $p^s$ and cardinality $p^{sm}$. In the present paper, we explore the construction of Cauchy MDS matrices over Galois rings. Moreover, we introduce a new approach that considers nilpotent elements and Teichm\"uller set of Galois ring $GR(p^s,p^{sm})$ to reduce the number of entries in these matrices. Furthermore, we construct $p^{(s-1)m}(p^m-1)$ distinct functions with the help of Frobenius automorphisms. These functions preserve MDS property of matrices. Finally, we prove some results using automorphisms and isomorphisms of the Galois rings that can be used to generate new Cauchy MDS matrices.