On groups generated by mixed type permutations and key addition groups

B. A. Pogorelov ; M. A. Pudovkina

Prikladnaya Diskretnaya Matematika. Supplement, no. 9 (2016), pp. 14-16.

Résumé

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.

Détail

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

Bibliographie

[1] Pogorelov B. A., Pudovkina M. A., “Nadgruppy additivnykh regulyarnykh grupp poryadka $2^n$ koltsa vychetov i vektornogo prostranstva”, Diskretnaya matematika, 27:3 (2015), 74–94 | DOI | MR

[2] Pogorelov B. A., Pudovkina M. A., “Orbitalnye proizvodnye po podgruppam i ikh kombinatorno-gruppovye svoistva”, Diskretnaya matematika, 27:4 (2015), 94–119 | DOI | MR

[3] Elizarov V. P., Konechnye koltsa, Gelios ARV, M., 2006