Generalisation of Hadamard matrix to generate involutory MDS matrices for lightweight cryptography

Abstract

In this study, the authors generalise Hadamard matrix over F-2m and propose a new form of Hadamard matrix, which they call generalised Hadamard (GHadamard) matrix. Then, they focus on generating lightweight (involutory) maximum distance separable (MDS) matrices. They also extend this idea to any k x k matrix form, where k is not necessarily a power of 2. The new matrix form, GHadamard matrix, is used to generate new 4 x 4 involutory MDS matrices over F-24 and F-28, and 8 x 8 involutory/non- involutory MDS matrices over F-24 by considering the minimum exclusive OR (XOR) count, which is a metric defined to estimate the hardware implementation cost. In this context, they improve the best-known results of XOR counts for 8 x 8 involutory/non-involutory MDS matrices over F-24.