Involutory-Multiple-Lightweight MDS Matrices based on Cauchy-type Matrices

One of the best methods for constructing maximum distance separable ( \begin{document}$ \operatorname{MDS} $\end{document} ) matrices is based on making use of Cauchy matrices. In this paper, by using some extensions of Cauchy matrices, we introduce several new forms of \begin{document}$ \operatorname{MDS} $\end{document} matrices over finite fields of characteristic 2. A known extension of a Cauchy matrix, called the Cauchy-like matrix, with application in coding theory was introduced in 1985. One of the main contributions of this paper is to apply Cauchy-like matrices to introduce \begin{document}$ 2n \times 2n $\end{document} involutory \begin{document}$ \operatorname{MDS} $\end{document} matrices in the semi-Hadamard form which is a generalization of the previously known methods. We make use of Cauchy-like matrices to construct multiple \begin{document}$ \operatorname{MDS} $\end{document} matrices which can be used in the Feistel structures. We also introduce a new extension of Cauchy matrices to be referred to as Cauchy-light matrices. The introduced Cauchy-light matrices are applied to construct \begin{document}$ n \times n $\end{document} \begin{document}$ \operatorname{MDS} $\end{document} matrices having at least \begin{document}$ 3n-3 $\end{document} entries equal to the unit element \begin{document}$ 1 $\end{document} ; such a matrix is called a lightweight \begin{document}$ \operatorname{MDS} $\end{document} matrix and can be used in the lightweight cryptography. A simple closed-form expression is given for the determinant of Cauchy-light matrices.