New concepts in design of lightweight MDS diffusion layers

Diffusion layers are an important part of most symmetric ciphers and MDS matrices can be used to construct perfect diffusion layers. However, there are few techniques for constructing these matrices with low implementation cost in software/hardware. In this article, we try to give some construction methods of MDS matrices with at least the following properties: Easy implementation, dynamic use and constructing a large family of MDS matrices from one (0, 1)-matrix which is a block-wise MDS matrix. For this purpose, we define an equivalence relation between rings and based on this definition, we classify and determine MDS matrices over equivalent rings. At first, we construct a new family of MDS matrices only with XORs and right or left shifts. Then, we construct another family of MDS matrices with XORs and cyclic shifts operations. Finally, we construct a family of lightweight MDS matrices with the same implementation cost as their inverses for the use in block ciphers.