On properties of the largest probability for difference transition under a random bijective group mapping

We consider two finite groups $(G_1,\otimes)$, $(G_2, \odot)$ with binary operations $ \otimes$, $\odot$. In practice, $G_1$ and $G_2$ are usually equal to the additive group $(V_m, \oplus)$ of the $m$-dimensional vector space $V_m$ over $\mathrm{GF}(2)$ or the additive group $(\mathbb{Z}_{2^m}, \boxplus)$ of the residues ring $\mathbb{Z}_{2^m}$. Nonabelian group of order $2^m$ having a cyclic subgroup of index $2$ can be considered as the nearest one to the additive group $(\mathbb{Z}_{2^m}, \boxplus)$. These groups are the dihedral group $(D_{2^{(m-1)}}, \diamond)$ and the generalized quaternion group $(Q_{2^m}, \boxtimes)$. In differential technique and its generalizations, each bijective mapping is associated with the differences table. In this paper, for all $\otimes, \odot \in \{\oplus, \boxplus, \boxtimes, \diamond \}$, we experimentally study a random value ${q^{( \otimes , \odot )}}$ that is equal to $|G_1|{p^{( \otimes , \odot )}}$, where ${p^{( \otimes , \odot )}}$ is the largest element of the differences table corresponding to a random mapping $s: G_1 \to G_2$. We consider randomly chosen bijective mappings as well as real S-boxes. As for all $\otimes, \odot \in \{\oplus, \boxplus, \boxtimes, \diamond \}$, we compute ${q^{( \otimes , \odot )}}$ for $S$-boxes of ciphers Aes, Anubis, Belt, Crypton, Fantomas, iScream, Kalyna, Khazad, Kuznyechik, Picaro, Safer, Scream, Zorro, Gift, Panda, Pride, Prince, Prost, Klein, Noekeon, Piccolo.