A note on cyclic MDS and non-MDS matrices

TL;DR

This paper investigates properties of cyclic and $g$-circulant matrices, establishing their relation to circulant matrices, analyzing their determinants, and showing limitations on their orthogonality and MDS properties over finite fields of characteristic 2.

Contribution

It establishes permutation equivalence between cyclic and circulant matrices, analyzes determinants of $g$-circulant matrices, and proves the impossibility of certain properties over fields of characteristic 2.

Findings

Cyclic matrices are permutation equivalent to circulant matrices.

Determinants of $g$-circulant matrices of order $2^d$ are determined.

Cyclic matrices cannot be both orthogonal and MDS over fields of characteristic 2.

Abstract

In $1998,$ Daemen {\it{ et al.}} introduced a circulant Maximum Distance Separable (MDS) matrix in the diffusion layer of the Rijndael block cipher, drawing significant attention to circulant MDS matrices. This block cipher is now universally acclaimed as the AES block cipher. In $2016,$ Liu and Sim introduced cyclic matrices by modifying the permutation of circulant matrices and established the existence of MDS property for orthogonal left-circulant matrices, a notable subclass within cyclic matrices. While circulant matrices have been well-studied in the literature, the properties of cyclic matrices are not. Back in $1961$, Friedman introduced $g$-circulant matrices which form a subclass of cyclic matrices. In this article, we first establish a permutation equivalence between a cyclic matrix and a circulant matrix. We explore properties of cyclic matrices similar to $g$-circulant matrices. Additionally, we determine the determinant of $g$-circulant matrices of order $2^d \times 2^d$ and prove that they cannot be simultaneously orthogonal and MDS over a finite field of characteristic $2$. Furthermore, we prove that this result holds for any cyclic matrix.