MDS matrices from skew polynomials with automorphisms and derivations
Atif Ahmad Khan, Shakir Ali, Elif Segah Oztas, Abhishek Kesarwani
TL;DR
This work constructs MDS diffusion matrices within skew polynomial rings $\mathbb{F}_q[X;\theta,\delta]$ by introducing $\delta_{\theta}$-circulant matrices, linking their entries to right multiplication by polynomials modulo $X^m-1$ and establishing a concrete MDS criterion in terms of polynomial weights. It then develops quasi recursive MDS matrices via companion-matrix products in the twisted setting, proving involutory quasi recursive MDS matrices under appropriate choices of $\theta$ and $\delta$, thus improving on prior quasi-involutory constructions. The paper also shows how Hadamard products generate many additional MDS matrices and provides illustrative examples, including explicit constructions over small finite fields. Open problems include a full characterization of MDS $\delta_{\theta}$-circulant matrices and potential extensions to orthogonal or semi-involutory variants.
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 $δ_θ$-circulant matrix can be viewed as a generalization of classical constructions obtained in the absence of $θ$-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 $\mathbb{F}_q[X;θ]$, 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.