On groups containing the additive group of the residue ring or the vector space

Groups which are most frequently used as key addition groups in iterative block ciphers include the regular permutation representation $V_n^ + $ of the group of vector key addition, the regular permutation representation $\mathbb{Z}_{{2^n}}^ + $ of the additive group of the residue ring, and the regular permutation representation $\mathbb{Z}_{{2^n} + 1}^ \odot $ of the multiplicative group of a prime field (in the case where ${2^n} + 1$ is a prime number). In this work we consider the extension of the group ${G_n}$ generated by $V_n^ + $ and $\mathbb{Z}_{{2^n}}^ + $ by means of transformations and groups which naturally arise in cryptographic applications. Examples of such transformations and groups are the groups $\mathbb{Z}_{{2^d}}^ + \times V_{n - d}^ + $ and $V_{n - d}^ + \times \mathbb{Z}_{{2^d}}^ + $ and pseudoinversion over the field $GF({2^n})$ or over the Galois ring $GR({2^{md}}{,2^m})$.