Development of Methods for Building Circulant MDS Matrices of Size n x n, 8 < n <= 16 With a Large Number of Units and a Small Number of Different Elements Over a Finite Field GF (q)

In this paper we research efficient MDS matrices, which allow cryptographic transformations to reach a required level of diffusion property. To construct them, in the beginning, we build bi-regular circulating patterns of size n x n, 9 ≤ n ≤ 20, with a large number of units and a small number of different elements. For the five patterns with n = 9, 10, 11 over a field of characteristic 2 we calculate a number of pairwise different minors and implement an appropriate method for MDS matrices building. Over the fields GF(2^k), k = 9, 10, 11, 12, 16, we test experimentally its efficiency. Finally, over the field GF(256) we propose a new method based on expansion of determinant in row allowing us to calculate matrix minors more efficiently and not to use pre-treatment. Also, it provides opportunity to deal with matrices of arbitrary patterns. Despite less running time of this method, comparing with the first one, getting stricter efficiency estimations requires an additional research.