On the Counting of Involutory MDS Matrices

The optimal branch number of MDS matrices has established their prominence in the design of diffusion layers for various block ciphers and hash functions. Consequently, several matrix structures have been proposed for designing MDS matrices, including Hadamard and circulant matrices. In this paper, we first provide the count of Hadamard MDS matrices of order $4$ over the field $\mathbb{F}_{2^r}$. Subsequently, we present the counts of order $2$ MDS matrices and order $2$ involutory MDS matrices over the field $\mathbb{F}_{2^r}$. Finally, leveraging these counts of order $2$ matrices, we derive an upper bound for the number of all involutory MDS matrices of order $4$ over $\mathbb{F}_{2^r}$.