Orthogonal MDS Diffusion Matrices over Galois Rings

Abstract

MDS matrices are important components in block cipher algorithm design, which provide diffusion of input bits. Recently, many constructions of MDS matrices focused on lightweight constructions. All MDS matrices constructions were over Galois field. In this paper, we give new construction of MDS matrices which is over Galois ring \(GR(2^n,k)=\mathbb {Z}_{2^n}[x]/(f(x))\), where f(x) is a basic irreducible polynomial of degree k over \(\mathbb {Z}_{2^n}\). We first construct Hadamard matrices over \(U(GR(2^n,k))\) by adding some signs on the entries of the matrices (i.e. performing entry-wise multiplication with enabling Hadamard \((1,-1)\)-matrices). We give complete enumerations of \(4 \times 4\) and \(8 \times 8\) enabling Hadamard \((1,-1)\)-matrices. We prove that there is no \(2 \times 2\) orthogonal MDS matrix over Galois ring \(GR(2^n,k)\) and construct \(4 \times 4\) orthogonal MDS matrices over \(GR(2^n,k)\).