On MDS Property of g-Circulant Matrices

• URL: http://arxiv.org/abs/2406.15872v1
• Date: Sat, 22 Jun 2024 15:18:31 GMT
• Title: On MDS Property of g-Circulant Matrices
• Authors: Tapas Chatterjee, Ayantika Laha,
• Abstract summary: We first discuss $g$-circulant matrices with involutory and MDS properties. We then delve into $g$-circulant semi-involutory and semi-orthogonal matrices with entries from finite fields.
• Score: 3.069335774032178
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $2022.$

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