Geodesic
$\otimes_{\mathbf W,\mathrm{ch}}$-markovian and imprimitive properties of block ciphers
Prikladnaya Diskretnaya Matematika. Supplement, no. 8 (2015), pp. 69-71.

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In this paper, we describe relations between $\otimes_{\mathbf W,\mathrm{ch}}$-markovian block ciphers and a wreath product. Let $X$ be an alphabet of plaintexts (ciphertexts) in iterated block ciphers, $(X,\otimes)$ be a regular abelian group, and $\mathbf W=\{W_0,\dots,W_{r-1}\}$ be a partition of $X$. In the case when $\mathbf W$ is the set of cosets of a subgroup of $(X,\otimes)$, we prove that $\otimes$-Markov block cipher is $\otimes_{\mathbf W,\mathrm{ch}}$-markovian iff $\mathbf W$ is an imprimitivity system of the group generated by round functions of the cipher. We show that there are $\otimes_{\mathbf W,\mathrm{ch}}$-markovian block ciphers where $\mathbf W$ is not a set of cosets. So, for the additive group $(V_n^+,\oplus)$ of the vector space $V_n$, we describe $\oplus_{\mathbf W,\mathrm{ch}}$-markovian classes of nonlinear and affine transformations for $\mathbf W$ being not a set of cosets. We show that the set of all affine $\oplus_{\mathbf W,\mathrm{ch}}$-markovian transformations on $V_n$ is a group and give examples of it.
Mots-clés : imprimitive group
Keywords: homomorphism method, XSL-block cipher, wreath product. 
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     author = {B. A. Pogorelov and M. A. Pudovkina},
     title = {$\otimes_{\mathbf W,\mathrm{ch}}$-markovian and imprimitive properties of block ciphers},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
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}
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B. A. Pogorelov; M. A. Pudovkina. $\otimes_{\mathbf W,\mathrm{ch}}$-markovian and imprimitive properties of block ciphers. Prikladnaya Diskretnaya Matematika. Supplement, no. 8 (2015), pp. 69-71. http://geodesic.mathdoc.fr/item/PDMA_2015_8_a24/

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