Classification of SPN Structures From the Viewpoint of Structural Cryptanalysis

In this paper, we provide two types of pattern matrices for the diffusion layer P, both can be used as classification criteria for the substitution-permutation network (SPN) structures: if the pattern matrices of distinct SPN structures are equal, then these structures may have the same impossible differential (ID)/zero correlation linear hull (ZC) and the same differential/linear active S-boxes. We introduce some interesting properties of the pattern matrices. Applying our results, we arrive at several interesting facts. First, all the SPN structures with MDS-type diffusion layers fall into the same class and have the same ID/ZC/minimum number of active S-boxes. Second, we provide several interesting properties of pattern matrices and build the links between the P-layer and that after several popular operations. Finally, we investigate the properties of pattern matrices for bit shuffles, the pattern matrices keep the same if and only if the n-partition characteristics of them are equal. Our results are helpful in the designing of block ciphers.