Cryptographic properties of orthomorphic permutations

In this paper, we consider bijective mappings F : Zn ^ Zn called orthomorphisms such that the mappings G(x) = F(x) ф x are also bijective. It is used in the Lai - Massey scheme as a mixing element between rounds and it also can be used to construct cryptographically strong S-boxes. The main cryptographic properties are studied, namely nonlinearity and differential uniformity. It was revealed that, for n = 2, 3, 4, the linear approximation tables of orthomorphisms consist of the values 0 and ±2^п-1, and the difference distribution tables consist of the values 0 and 2ⁿ. It turned out that or-thomorphisms of a small number of variables are not resistant to linear and differential cryptanalysis.

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