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

B. A. Pogorelov ; M. A. Pudovkina

Diskretnaya Matematika, Tome 28 (2016) no. 4, pp. 100-121

Résumé

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 $GR({2^{md}}{,2^m})$.

Détail

Keywords: key addition group, additive regular group, wreath product of permutation groups, multiplicative group of the residue ring, Galois ring.