Geodesic
On properties of $W$-permutations over the residue ring
Prikladnaya Diskretnaya Matematika. Supplement, no. 10 (2017), pp. 92-93.

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From the theory of Markov chains, it follows that the states of a Markov chain can be enlarged by a partition $mathbf W$ of $\mathbb Z_n$ if a number of conditions are satisfied for the blocks of this partition and the elements of the difference table of a permutation on $\mathbb Z_n$. However, using a modification of the differential attack, we require a condition for only one block $W$ of the partition $\mathbf W$. In this case, the permutations satisfying such a “softened” requirement for the block $W$ are called $W$-permutations. Their properties are studied and described in this paper.
Keywords: Markov block ciphers, differential attack.
Mots-clés : enlargement of Markov chain, $W$-permutation 
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M. A. Pudovkina; A. S. Makeyev. On properties of $W$-permutations over the residue ring. Prikladnaya Diskretnaya Matematika. Supplement, no. 10 (2017), pp. 92-93. http://geodesic.mathdoc.fr/item/PDMA_2017_10_a36/

[1] Pogorelov B. A., Pudovkina M. A., “$\otimes_{\mathbf W,\mathrm{ch}}$-markovskie preobrazovaniya”, Prikladnaya diskretnaya matematika. Prilozhenie, 2015, no. 8, 17–19

[2] Kemeni D., Snell D., Konechnye tsepi Markova, Nauka, M., 1970 | MR