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 Prikl. Diskr. Mat. Suppl., 2016, Issue 9, Pages 14–16 (Mi pdma290)

Theoretical Foundations of Applied Discrete Mathematics

On groups generated by mixed type permutations and key addition groups

B. A. Pogorelova, M. A. Pudovkinab

a Academy of Cryptography of Russian Federation, Moscow
b National Engineering Physics Institute (MEPhI), Moscow

Abstract: 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.

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

DOI: https://doi.org/10.17223/2226308X/9/5

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Document Type: Article
UDC: 519.7

Citation: B. A. Pogorelov, M. A. Pudovkina, “On groups generated by mixed type permutations and key addition groups”, Prikl. Diskr. Mat. Suppl., 2016, no. 9, 14–16

Citation in format AMSBIB
\Bibitem{PogPud16} \by B.~A.~Pogorelov, M.~A.~Pudovkina \paper On groups generated by mixed type permutations and key addition groups \jour Prikl. Diskr. Mat. Suppl. \yr 2016 \issue 9 \pages 14--16 \mathnet{http://mi.mathnet.ru/pdma290} \crossref{https://doi.org/10.17223/2226308X/9/5}