On the Counting of Involutory MDS Matrices

The optimal branch number of MDS matrices has established their importance in designing diffusion layers for various block ciphers and hash functions. As a result, numerous matrix structures, including Hadamard and circulant matrices, have been proposed for constructing MDS matrices. Also, in the literature, significant attention is typically given to identifying MDS candidates with optimal implementations or proposing new constructions across different orders. However, this paper takes a different approach by not emphasizing efficiency issues or introducing novel constructions. Instead, its primary objective is to enumerate Hadamard MDS and involutory Hadamard MDS matrices of order $4$ within the field $\mathbb{F}_{2^r}$. Specifically, it provides an explicit formula for the count of both Hadamard MDS and involutory Hadamard MDS matrices of order $4$ over $\mathbb{F}_{2^r}$. Additionally, the paper presents the counts of order $2$ MDS matrices and order $2$ involutory MDS matrices over $\mathbb{F}_{2^r}$. Finally, leveraging these counts of order $2$ matrices, an upper bound is derived for the number of all involutory MDS matrices of order $4$ over $\mathbb{F}_{2^r}$.