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

2025/12/22 by Shakir Ali, Ali, Shakir, Atif Ahmad Khan +3
Mathematics · Engineering · Computer Science · #Rings, Modules, and Algebras #graph theory and CDMA systems #Digital Image Processing Techniques

paper · doi:10.48550/arxiv.2512.19306

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 Teichmüller 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.

Citations

• Exhaustive Search for Small Dimension Recursive MDS Diffusion Layers for Block Ciphers and Hash Functions

Related

• Quasi-recursive MDS Matrices over Galois Rings
• Bases of the Galois Ring GR(p^r,m) over the Integer Ring Z_p^r
• Slope filtrations for relative Frobenius
• Diagonalizing the Frobenius
• Constructing semisimple p-adic Galois representations with prescribed properties