⊗W, _ch-markovian and imprimitive properties of block ciphers

In this paper, we describe relations between ® _W)Ch-markovian block ciphers and a wreath product. Let X be an alphabet of plaintexts (ciphertexts) in iterated block ciphers, (X, ®) be a regular abelian group, and W = {W ₀,..., W _r-1} be a partition of X. In the case when W is the set of cosets of a subgroup of (X, ®), we prove that ®-Markov block cipher is ® _W;Ch-markovian iff W is an imprimitivity system of the group generated by round functions of the cipher. We show that there are ® _W;Ch-markovian block ciphers where W is not a set of cosets. So, for the additive group (V+, Ф) of the vector space Vn, we describe ф-w^h-markovian classes of nonlinear and affine transformations for W being not a set of cosets. We show that the set of all affine ф-^h-markovian transformations on V _n is a group and give examples of it.

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