MDS matrices from skew polynomials with automorphisms and derivations

2026-02-01

AI Summary

This work addresses the need for efficient and secure linear transformations in symmetric cryptography and coding theory by investigating constructions of MDS matrices with optimal diffusion properties. Building upon the skew polynomial ring $\mathbb{F}_q[X;\theta,\delta]$ equipped with an automorphism $\theta$ and a $\theta$-derivation $\delta$, the study introduces, for the first time, a $\theta$-derivation mechanism to define novel $\delta_\theta$-circulant matrices and quasi-recursive MDS matrices. It rigorously derives necessary and sufficient conditions for these matrices to satisfy both the MDS property and involutory characteristics. The resulting constructions achieve strict involutory behavior alongside optimal diffusion, offering structural advantages over existing quasi-involutive approaches. This work thus extends the classical framework for MDS matrix construction, with multiple concrete instances demonstrating the effectiveness and practicality of the proposed method.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Maximum Distance Separable (MDS) matrices play a central role in coding theory and symmetric-key cryptography due to their optimal diffusion properties. In this paper, we present a construction of MDS matrices using skew polynomial rings \( \mathbb{F}_q[X;\theta,\delta] \), where \( \theta \) is an automorphism and \( \delta \) is a \( \theta\)-derivation on \( \mathbb{F}_q \). We introduce the notion of \( \delta_{\theta} \)-circulant matrices and study their structural properties. Necessary and sufficient conditions are derived under which these matrices are involutory and satisfy the MDS property. The resulting $\delta_\theta$-circulant matrix can be viewed as a generalization of classical constructions obtained in the absence of $\theta$-derivations. One of the main contribution of this work is the construction of quasi recursive MDS matrices. In the setting of the skew polynomial ring $\mathbb{F}_q[X;\theta]$, we construct quasi recursive MDS matrices associated with companion matrices. These matrices are shown to be involutory, yielding a strict improvement over the quasi-involutory constructions previously reported in the literature. Several illustrative results and examples are also provided.

Problem

• MDS matrices
• skew polynomials
• automorphisms
• derivations
• involutory matrices

Innovation

• skew polynomial rings
• MDS matrices
• θ-derivation
• δ_θ-circulant matrices
• quasi recursive