Construction of Hadamard-based MixColumns Matrices Resistant to Related-Differential Cryptanalysis

In this paper, we study MDS matrices that are specifically designed to prevent the occurrence of related differentials. We investigate MDS matrices with a Hadamard structure and demonstrate that it is possible to construct 4 X 4 Hadamard matrices that effectively eliminate related differentials. Incorporating these matrices into the linear layer of AES-like block-ciphers/hash functions significantly mitigates the attacks that exploit the related differentials property. The central contribution of this paper is to identify crucial underlying relations that determine whether a given 4 X 4 Hadamard matrix exhibits related differentials. By satisfying these relations, the matrix ensures the presence of related differentials, whereas failing to meet them leads to the absence of such differentials. This offers effective mitigation of recently reported attacks on reduced-round AES. Furthermore, we propose a faster search technique to exhaustively verify the presence or absence of related differentials in 8 X 8 Hadamard matrices over finite field of characteristic 2 which requires checking only a subset of involutory matrices in the set. Although most existing studies on constructing MDS matrices primarily focus on lightweight hardware/software implementations, our research additionally introduces a novel perspective by emphasizing the importance of MDS matrix construction in relation to their resistance against differential cryptanalysis.