On groups generated by mixed type permutations and key addition groups

Three groups are often used as key addition groups in iterated block ciphers: $V_n^+$, $\mathbb Z_{2^n}^+$ and $\mathbb Z_{2^n+1}^\odot$. They are the regular permutation representations, respectively, of the group of vector key addition, of the additive group of the residue ring $\mathbb Z_{2^n}$, and of the multiplicative group of the residue ring $\mathbb Z_{2^n+1}$, where $2^n+1$ is a prime number. In this paper, we describe some properties of the extensions of the group ${G_n}=\langle V_n^+,\mathbb Z_{2^n}^+\rangle$ by transformations and groups related to cryptographic applications. The groups $\mathbb Z_{2^d}^+ \times V_{n-d}^+$, $V_{n-d}^+\times\mathbb Z_{2^d}^+$ and a pseudoinverse permutation of the field $\operatorname{GF}(2^n)$ or the Galois ring $\operatorname{GR}(2^{md},2^m)$ are examples of such groups and transformations.