New Directions in the Design of Binary Matrices for SPN Block Ciphers

The diffusion layer plays an important role in a block cipher. Some block ciphers, such as ARIA, Camellia, and Skinny use binary matrices as diffusion layers which can be efficiently implemented in hardware and software. In this paper, the goal is to propose some new binary matrices with suitable values for the active S-boxes for R rounds. Firstly, some new $16 \times 16$ matrices are proposed whose software implementations are better than the corresponding one for the ARIA block cipher. Also, the values for the minimum active S-boxes for these matrices are greater than the corresponding values for the ARIA block cipher for $R>5$.To design $32 \times 32$ matrices, a structure with a special form is proposed. Using this structure, a $32\times 32$ binary matrix is proposed which guarantees at least 48 active S-boxes for 8 rounds of an SPN structure with this matrix as its diffusion layer. By extending this structure, a $32\times 32$ non-binary matrix is presented which results in at least 60 active S-boxes after 8 rounds.