Construction of $4\times 4$ MDS Matrices for Lightweight Block Ciphers

Diffusion is one of the fundamental properties while designing block ciphers. Diffusion layer is derived with a linear diffusion matrix which converts an i/p vector to o/p vectors by the diffusion operations. To resist the differential and linear attacks, it is indispensable to strengthen the diffusion power of the matrix for a block cipher. Maximizing the branch number is certainly helpful to obtain the stronger security requirements along with better computational performance. The matrix having maximum branch number is an ideal and most suitable diffusion layer. We call this matrix $\mathcal{M}$ aximal $\mathcal{D}$ istance Separable (MDS) matrix. MDS matrices are important constituent for block ciphers and are widely used in various ciphers such as AES [1], LED [2], SKINNY [3] etc. It is essential to minimize the computational complexity and implementation costs while constructing the diffusion layer. This paper presents an algorithm to construct $4\times 4$ dimension matrices. Utilization of some specific structure of matrices and minimal XOR counts construct these matrices as lightweight in nature. The practical adaptability and optimality of the proposed approach is also discussed through various experimental evaluations.