Geodesic
Nonabelian key addition groups and $\otimes _{\mathbf{W}}$-markovian property of block ciphers
Matematičeskie voprosy kriptografii, Tome 11 (2020) no. 4, pp. 107-131 Cet article a éte moissonné depuis la source Math-Net.Ru

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For an Abelian key addition group $\left( {X, \otimes } \right)$ and a partition ${\bf{W}} = \{ {W_0},\ldots ,{W_{r-1}}\}$ of a set $X$ we had introduced ${ \otimes _{\bf{W}}}$-markovian transformations and ${ \otimes _{\bf{W}}}$-markovian ciphers. The ${ \otimes _{\bf{W}}}$-markovian condition is required to validate different generalizations of differential technique. In this paper, we study ${ \otimes _{\bf{W}}}$-markovian ciphers and transformations on an nonabelian group $\left( {X, \otimes } \right)$. We get restrictions on the structure of groups $(X, \otimes )$, $\left\langle {{g_k}|k \in X} \right\rangle $ and blocks of a nontrivial partition ${\bf{W}}$ as a consequence of the condition of partial preservation of $\bf{W}$ by the round function ${g_k}\colon X \to X$ for all $k \in X$. For all nonabelian groups of the order ${2^m}$ with a cyclic subgroup having index $2$ we describe classes of ${ \otimes _{\bf{W}}}$-markovian permutations. 
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B. A. Pogorelov; M. A. Pudovkina. Nonabelian key addition groups and $\otimes _{\mathbf{W}}$-markovian property of block ciphers. Matematičeskie voprosy kriptografii, Tome 11 (2020) no. 4, pp. 107-131. http://geodesic.mathdoc.fr/item/MVK_2020_11_4_a6/

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