Construction and Filtration of Lightweight Formalized MDS Matrices

Zhang Shi-Yi, Wang Yong-juan, Gao Yang, Wang Tao Corresponding author: Wang Yong-juan, E-mail: [email protected] Abstract: The 4 4  MDS matrix over 2 F is widely used in the design of block cipher's linear diffusion layers. However, considering the cost of a lightweight cipher's implementation, the sum of XOR operations of a MDS matrix usually plays the role of measure. During the research on the construction of the lightweight 4 4  MDS matrices, this paper presents the concept of formalized MDS matrix: some of the entries that make up the matrix are known, and their positions are determined, and the criterions of the MDS matrix is satisfied. In this paper, using the period and minimal polynomial theory of entries over finite fields, a new construction method of formalized MDS matrices is proposed. A large number of MDS matrices can be obtained efficiently by this method, and their number distribution has significant structural features. However, the algebraic structure of the lightest MDS matrices is also obvious. This paper firstly investigates the construction of 4 4  lightweight MDS matrices, analyzes the distribution characteristics of the them, and the feasibility of the construction method. Then, for the lightest MDS matrices obtained from the method above, the algebraic relations in themselves and between each other are studied, and the important application of the alternating group 4 A and it's subgroup, the Klein four-group is found.