In symmetric cryptography, maximum distance separable (MDS) matrices with computationally simple inverses have wide applications. Many block ciphers like AES, SQUARE, SHARK, and hash functions like PHOTON use an MDS matrix in the diffusion layer. In this article, we first characterize all $3 \times 3$ irreducible semi-involutory matrices over the finite field of characteristic $2$. Using this matrix characterization, we provide a necessary and sufficient condition to construct MDS semi-involutory matrices using only their diagonal entries and the entries of an associated diagonal matrix. Finally, we count the number of $3 \times 3$ semi-involutory MDS matrices over any finite field of characteristic $2$.

翻译：在对称密码学中，具有计算简单逆的最大距离可分（MDS）矩阵具有广泛的应用。许多分组密码（如AES、SQUARE、SHARK）和哈希函数（如PHOTON）在扩散层中均使用MDS矩阵。本文首先刻画了特征为2的有限域上所有3×3不可约半对合矩阵。利用这一矩阵刻画，我们给出了仅使用其对角线元素及一个关联对角矩阵的元素来构造MDS半对合矩阵的充分必要条件。最后，我们统计了特征为2的任意有限域上3×3半对合MDS矩阵的数量。