MDS matrices from skew polynomials with automorphisms and derivations

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 \( \mathbb{F}_q[X;θ,δ] \), where \( θ\) is an automorphism and \( δ\) is a \( θ\)-derivation on \( \mathbb{F}_q \). We introduce the notion of \( δ_θ \)-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 $\hat{I}{´}_{\hat{I}}¸$-circulant matrix can be viewed as a generalization of classical constructions obtained in the absence of $\hat{I}¸$-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 ${\mathbb{F}}_q[X;\hat{I}¸]$, 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.