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

2025-12-22

AI Summary

Lightweight cryptography demands efficient and secure MDS diffusion layers, particularly structured Cauchy-type MDS matrices over Galois rings of characteristic $p^s$. Method: This work systematically constructs Cauchy-type MDS matrices over such rings by innovatively integrating nilpotent elements with the Teichmüller set to simplify element representation; employs the Frobenius endomorphism to generate an explicit family of $p^{(s-1)m}(p^m-1)$ MDS-preserving functions; and derives scalable new matrix families via ring automorphisms and isomorphisms. Contribution/Results: All constructed matrices are rigorously proven to satisfy the MDS property. The proposed framework drastically reduces parameter search complexity and achieves, for the first time, the batch generation of structured, provably secure, and computationally efficient Cauchy MDS matrices over Galois rings.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Distributed Search
• Cognitive Modeling & Cognitive Systems: (Computational) Cognitive Architectures

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

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.

Problem

• Construct Cauchy MDS matrices over Galois rings using nilpotent elements
• Reduce matrix entries via Teichmüller sets and Frobenius automorphisms
• Generate new MDS matrices with ring automorphisms and isomorphisms

Innovation

• Constructs Cauchy MDS matrices using nilpotent elements and Teichmüller sets
• Reduces matrix entries via Galois ring properties and Frobenius automorphisms
• Generates new MDS matrices through automorphisms and isomorphisms of Galois rings