Perfect involutory diffusion layers based on invertibility of some linear functions

One of the most important structures used in modern block ciphers is the substitution-permutation network (SPN) structure. Many block ciphers with this structure widely use Maximun Distance Separable (MDS) matrices over finite fields as their diffusion layers, for example, advanced encryption standard (AES) uses a 4-4 MDS matrix as the main part of its diffusion layer and the block cipher Khazad has an involutory 8-8 matrix. In this study, first a construction is proposed for a 4-4 linear diffusion layer that can intermix four words of arbitrary size with branch number 5. Then extend this idea for 8-8 diffusion layer using low-cost linear functions. In this construction, first, certain binary linear combinations of inputs are fed into two or three different invertible linear functions and then combined using XOR operation. In order to show the efficiency of the proposed diffusion layer, the authors exploit it in a nested SPN structure and compare its efficiency with some well-known diffusion layers such as the diffusion layer of Hierocrypt.