🤖 AI Summary
This work addresses the construction and enumeration of $3 imes 3$ irreducible, semi-self-inverse MDS matrices over finite fields of characteristic two. Motivated by the cryptographic requirement for diffusion layers that simultaneously achieve optimal diffusion (MDS property) and computational efficiency (semi-self-inverseness), we provide the first complete algebraic characterization of such matrices: we prove that their necessary and sufficient conditions depend solely on the diagonal entries and the associated diagonal matrix, and we derive a constructive criterion grounded in finite-field algebra and the MDS determinant condition. Furthermore, via rigorous enumeration and parametric analysis over arbitrary characteristic-two finite fields $mathbb{F}_{2^m}$, we obtain an exact closed-form counting formula for all such matrices. The results establish a theoretical foundation and a verifiable, algebraically guided construction framework for lightweight block cipher design.
📝 Abstract
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 $3 imes 3$ 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 $3 imes 3$ semi-involutory MDS matrices over any finite field of characteristic $2$.