The Design of Unified Structures with 4/8 Branches

Recently, Liu et al. and Sun et al. proposed the unified structure to describe structures that share similar procedures for decryption and encryption. Furthermore, Sun et al. explored the regular unified structure and gave its \mathscrX-type normalized form. This paper focuses on the selection of parameters of the \mathscrX-type normalized form with 4 or 8 branches, where parameter matrices are composed by either invertible matrices or the zero matrix. The main results are as follows: First, we show that the dual structure of any structure in the \mathscrX-type normalized form is affine equivalent to its decryption structure, and furthermore identify critical conditions to rule out some insecure instances of unified structures. Then, we relax the condition on the number of the full-diffusion round of unified structures to be 2d-2 which was shown at CRYPTO 2024. For example, the number of the full-diffusion round of the Type-I Generalized Feistel Structure, which was not determined previously, is equal to that of the Mars-like structure. Finally, we construct some instances of unified structures with 4 or 8 branches, along with the cryptanalysis of these structures.