On MDS Property of g-Circulant Matrices

TL;DR

This paper investigates the MDS and involutory properties of g-circulant matrices, establishing conditions under which these matrices can be MDS and exploring their semi-involutory and semi-orthogonal variants over finite fields.

Contribution

It extends prior results on circulant matrices by analyzing g-circulant matrices' MDS and involutory properties, providing new conditions and properties for these matrices.

Findings

g-circulant involutory matrices are MDS only if g ≡ -1 mod k

k-th power of associated diagonal matrices yields scalar matrices

extends previous circulant matrix results to g-circulant matrices

Abstract

Circulant Maximum Distance Separable (MDS) matrices have gained significant importance due to their applications in the diffusion layer of the AES block cipher. In $2013$, Gupta and Ray established that circulant involutory matrices of order greater than $3$ cannot be MDS. This finding prompted a generalization of circulant matrices and the involutory property of matrices by various authors. In $2016$, Liu and Sim introduced cyclic matrices by changing the permutation of circulant matrices. In $1961,$ Friedman introduced $g$-circulant matrices which form a subclass of cyclic matrices. In this article, we first discuss $g$-circulant matrices with involutory and MDS properties. We prove that $g$-circulant involutory matrices of order $k \times k$ cannot be MDS unless $g \equiv -1 \pmod k.$ Next, we delve into $g$-circulant semi-involutory and semi-orthogonal matrices with entries from finite fields. We establish that the $k$-th power of the associated diagonal matrices of a $g$-circulant semi-orthogonal (semi-involutory) matrix of order $k \times k$ results in a scalar matrix. These findings can be viewed as an extension of the results concerning circulant matrices established by Chatterjee {\it{et al.}} in $2022.$