New Insights into Involutory and Orthogonal MDS Matrices
Yogesh Kumar, Susanta Samanta, Atul Gaur
TL;DR
The paper investigates the structural relationships among MDS matrices under involutory, orthogonal, semi-involutory, and semi-orthogonal constraints over the finite field $\mathbb{F}_{2^m}$. It introduces a representative matrix form $M_1$ and a decomposition $M = \Phi(D_1,D_2,M_1)$ to link counts across these classes, enabling bidirectional enumeration between involutory and semi-involutory, and between orthogonal and semi-orthogonal matrices. The authors derive a closed-form count for $3\times3$ orthogonal MDS matrices as $(2^m-2)(2^m-3)(2^m-4)$ and provide corresponding formulas for $3\times3$ semi-involutory and semi-orthogonal matrices, as well as a joint $3\times3$ class count. They also characterize the general structure of $n\times n$ orthogonal matrices over $\mathbb{F}_{2^m}$, derive a closed form for $3\times3$ orthogonal MDS matrices, and extend enumeration to $4\times4$ matrices for $m=3$ to $8$, with practical implications for diffusion layers in block ciphers and hash functions. Overall, the work offers algebraic bridges between matrix classes, enabling efficient counting and construction insights for diffusion-dominant cryptographic primitives.
Abstract
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. 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 $\mathbb{F}_{2^m}$. Based on this generic form, we provide a closed-form expression for enumerating all $3 \times 3$ orthogonal MDS matrices over $\mathbb{F}_{2^m}$. Finally, leveraging the aforementioned interconnections, we present explicit formulas for counting $3 \times 3$ semi-involutory MDS matrices and semi-orthogonal MDS matrices.