Constructions of Maximal Distance Separable Matrices with Minimum XOR-counts

With the development of the internet of things, small-scale communication devices such as wireless sensors and the Radio Frequency IDentification(RFID) tags are widely used, these micro-devices have limited computing power, so that the traditional cryptographic algorithms are difficult to implement on these devices. How to construct a high-efficiency diffusion layer becomes an urgent problem. With the best diffusion property, the Maximal Distance Separable (MDS) matrix is often used to construct the diffusion layer of block ciphers. The number of XOR operations (XORs) is an indicator of the efficiency of hardware applications. Combined with the XORs calculation method which can evaluate hardware efficiency more accurately and a matrix with special structure——Toeplitz matrix, efficient MDS matrices with less XORs can be constructed. Using the structural characteristics of the Toeplitz matrix, the constraints of matrix elements are improved, and the complexity of matrices searching is reduced. The 4×4 MDS matrices and the 6×6 MDS matrices with the least XORs in the finite field ${\mathbb{F}_{{2^8}}}$ are obtained, and the 5×5 MDS matrices with the XORs which are equal to the known optimal results are obtained too. The proposed method of constructing MDS Toeplitz matrices with the least XORs has significance on the design of lightweight cryptographic algorithms.