MDS matrices from skew polynomials with automorphisms and derivations
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arXiv
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| Format: | Preprint |
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2026
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| author | Khan, Atif Ahmad Ali, Shakir Oztas, Elif Segah Kesarwani, Abhishek |
| author_facet | Khan, Atif Ahmad Ali, Shakir Oztas, Elif Segah Kesarwani, Abhishek |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_01383 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | MDS matrices from skew polynomials with automorphisms and derivations Khan, Atif Ahmad Ali, Shakir Oztas, Elif Segah Kesarwani, Abhishek Information Theory 94A60, 15A99, 11T06, 16S36 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. |
| title | MDS matrices from skew polynomials with automorphisms and derivations |
| topic | Information Theory 94A60, 15A99, 11T06, 16S36 |
| url | https://arxiv.org/abs/2602.01383 |