Construction of New Hadamard Matrix Forms to Generate 4X4 and 8X8 Involutory MDS Matrices over GF (2m) for Lightweight Cryptography

Abstract

In this paper, we present the construction of two Hadamard matrix forms over GF(2m) to generate 4×4 and 8×8 involutory MDS (IMDS) matrices. The first form provides a straightforward way to generate 4×4 IMDS matrices,
while the second is an efficient way to generate 8×8 IMDS matrices using a hybrid (combination of search-based methods and direct construction) approach. In addition, we propose an algorithm for computing the branch number of any non-singular matrix over GF(2m) and improve its computational complexity for Hadamard matrices. Using this algorithm and the proposed Hadamard matrix form, we obtain 2k ×2k lightweight involutory and non-involutory Hadamard MDS matrices with low XOR counts for k=2,3. Finally, we carry out a comparative study based on the XOR count to demonstrate that MDS matrices created using our Hadamard matrix forms have lower XOR counts than MDS matrices available in the literature as of today.