On the Construction of the 4 x 4 Lightest Circulant MDS Matrices

Abstract

4 x 4 MDS (Maximal Distance Separable) matrices with few XORs have a wide range of applications in many mainstream lightweight ciphers. For 4 x 4 circulant MDS matrices over GL(4,F2), they have at least 12 XOR operations. In this paper, by traversing their structure characteristics, the utter construction and the numeration of the lightest circulant MDS matrices are firstly investigated. Then the overall structure and the diagrams of these matrices are given. Finally the characteristics of permutation group in the lightest circulant MDS matrices above are found: they possess characteristics of symmetric group S4, and for a kind of particular MDS matrices, they can even form a Klein four-group in some ways.