A note on semi-orthogonal (G-matrix) and semi-involutory MDS matrices

Abstract

Abstract: Maximum Distance Separable (MDS) matrices are widely used in various cryptographic constructions since they provide perfect diffusion. Further, MDS matrices with easy-to-implement inverses are useful in designing diffusion layers in block ciphers. It is known that the inverse of an MDS matrix is computationally inexpensive if the matrix is either orthogonal or involutory. Generalizing the notion of orthogonal matrices, Fiedler et al. introduced semi-orthogonal property in 2012. Following this, Cheon et al. introduced semi-involutory property to generalize the involutory property in 2021. In both these cases, the aim of the authors was to construct matrices having computationally simple inverses.

In this work, we show that some existing Cauchy and Vandermonde based constructions of MDS matrices satisfy semi-orthogonal properties. We give some characterization of and semi-involutory and semi-orthogonal matrices in light of the MDS property. We also provide some results on circulant matrices with semi-involutory and semi-orthogonal properties. Finally we give a characterization of semi-involutory matrices which is a generalization of the case of Cheon et al.