On properties of $W$-permutations over the residue ring

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.