A Characterization of Semi-Involutory MDS Matrices

TL;DR

This paper characterizes all 3x3 semi-involutory MDS matrices over finite fields of characteristic 2, providing conditions for their construction and counting their number, which is valuable for cryptographic applications.

Contribution

It offers a complete characterization of semi-involutory MDS matrices and a method to construct and count them over finite fields of characteristic 2.

Findings

Characterization of all 3x3 irreducible semi-involutory matrices over GF(2)

Necessary and sufficient conditions for constructing MDS semi-involutory matrices

Counting the number of such matrices over finite fields of characteristic 2

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 $3 \times 3$ 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 $2$.