Almost Orthogonal MDS Matrices over GR(2n, k)

Abstract

MDS matrices are excellent candidate for providing diffusion properties in block ciphers. While MDS matrices over $\mathbb{F}_{2^{k}}$ are more commonly used, the paper [12] suggested to consider the use of MDS matrices over Galois ring $GR(2^{n}, k)$ . In this paper, we explore some constructions of MDS matrices over $GR(2^{n}, k)$ with special properties (e.g, Hadamard, almost orthogonal) from given MDS matrices over $\mathbb{F}_{2^{k}}$ . We show that $2^{r}\times 2^{r}$ enabling Hadamard (1, −1) -matrices do not exist for $r\geq 4$ . We also give an alaorithm to construct almost orthogonal MDS matrices over $GR({2}^{n}, k)$ .