MDS matrices play a critical role in the design of diffusion layers for block ciphers and hash functions due to their optimal branch number. Involutory and orthogonal MDS matrices offer additional benefits by allowing identical or nearly identical circuitry for both encryption and decryption, leading to equivalent implementation costs for both processes. These properties have been further generalized through the notions of semi-involutory and semi-orthogonal matrices. While much of the existing literature focuses on identifying efficiently implementable MDS candidates or proposing new constructions for MDS matrices of various orders, this work takes a different direction. Rather than introducing novel constructions or prioritizing implementation efficiency, we investigate structural relationships between the generalized variants and their conventional counterparts. Specifically, we establish nontrivial interconnections between semi-involutory and involutory matrices, as well as between semi-orthogonal and orthogonal matrices. Exploiting these relationships, we show that the number of semi-involutory MDS matrices can be directly derived from the number of involutory MDS matrices, and vice versa. A similar correspondence holds for semi-orthogonal and orthogonal MDS matrices. We also examine the intersection of these classes and show that the number of $3 \times 3$ MDS matrices that are both semi-involutory and semi-orthogonal coincides with the number of semi-involutory MDS matrices over $\mathbb{F}_{2^m}$. Furthermore, we derive the general structure of orthogonal matrices of arbitrary order $n$ over $\mathbb{F}_{2^m}$. Finally, leveraging the aforementioned interconnections, we present an alternative and direct derivation of the explicit formulae for counting $3 \times 3$ semi-involutory MDS matrices and $3 \times 3$ semi-orthogonal MDS matrices.

翻译：MDS矩阵因其最优分支数而在分组密码与哈希函数的扩散层设计中起着关键作用。对合与正交MDS矩阵通过允许加解密过程使用相同或近乎相同的电路结构，使得两个过程的实现成本等价，从而提供了额外优势。这些性质已通过对合矩阵与半正交矩阵的概念得到进一步推广。现有文献大多聚焦于识别可高效实现的MDS候选矩阵，或为不同阶数的MDS矩阵提出新构造方法，而本研究则采取了不同方向。我们并非引入新颖构造或优先考虑实现效率，而是探究广义变体与其传统对应形式之间的结构关系。具体而言，我们建立了半对合矩阵与对合矩阵之间，以及半正交矩阵与正交矩阵之间的非平凡相互联系。利用这些关系，我们证明半对合MDS矩阵的数量可直接从对合MDS矩阵的数量推导得出，反之亦然。半正交与正交MDS矩阵之间也存在类似对应关系。我们还考察了这些类别的交集，证明同时满足半对合与半正交性质的 $3 \times 3$ MDS矩阵数量，与 $\mathbb{F}_{2^m}$ 上半对合MDS矩阵的数量一致。此外，我们推导了 $\mathbb{F}_{2^m}$ 上任意阶数 $n$ 的正交矩阵的通用结构。最后，基于前述相互联系，我们提出了对 $3 \times 3$ 半对合MDS矩阵与 $3 \times 3$ 半正交MDS矩阵进行显式计数的替代性直接推导方法。