On Constructions of a Sort of MDS Block Diffusion Matrices for Block Ciphers and Hash Functions

Many modern block ciphers use maximum distance separate (MDS) matrices as their diffusion layers. In this paper, we propose a new method to verify a sort of MDS diffusion block matrices whose blocks are all polynomials in a certain primitive block over the finite field F2. And then we discover a new kind of transformations that can retain MDS property of diffusion matrices and generate a series of new MDS matrices from a given one. Moreover, we get an equivalence relation from this kind of transformation. And MDS property is an invariant with respect to this equivalence relation which can greatly reduce the amount of computation when we search for MDS matrices. The minimal polynomials of matrices play an important role in our strategy. To avoid being too theoretical, we list a series of MDS diffusion matrices obtained from our method for some specific parameters. Furthermore, we talk about MDS recursive diffusion layers with our method and extend the corresponding work of M. Sajadieh et al. published on FSE 2012 and the work of S. Wu published on SAC 2012.