MDS matrices from skew polynomials with automorphisms and derivations

TL;DR

This paper introduces a novel method for constructing MDS matrices using skew polynomial rings with automorphisms and derivations, leading to quasi recursive, involutory matrices with improved properties.

Contribution

It develops a new framework for MDS matrix construction via skew polynomials with automorphisms and derivations, including the concept of elta_{ heta}-circulant matrices and their properties.

Findings

Constructed quasi recursive MDS matrices with involutory property.

Established necessary and sufficient conditions for elta_{ heta}-circulant matrices to be MDS.

Demonstrated the superiority of involutory matrices over previous quasi-involutory constructions.

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.