$\otimes_{\mathbf W,\mathrm{ch}}$-markovian transformations

Let $X$ be an alphabet of plaintexts (ciphertexts) of iterated block ciphers and $(X,\otimes)$ be a regular abelian group. The group operation $\otimes$ defines the difference of a text pair. $\otimes$-Markov ciphers are defined as iterated ciphers of which round functions satisfy the condition that the differential probability is independent of the choice of plaintexts from $X$. For $\otimes$-Markov ciphers with independent round keys, the sequence of round differences forms a Markov chain. In this paper, we consider $\otimes$-Markov ciphers and a partition $\mathbf W=\{W_0,\dots,W_{r-1}\}$ with blocks being lumped states of the Markov chain. An $l$-round $\otimes$-Markov cipher is called $\otimes_{\mathbf W,\mathrm{ch}}$-markovian if the cipher and $\mathbf W$ satisfy the following condition: the block numbers sequence $j_0,\dots,j_l$ such that, for all $i\in\{0,\dots,l\}$, the $i^{th}$-round difference belongs to $W_{j_i}$ is a Markov chain. This definition can be also extended for permutations on $X$. For a partition $\mathbf W$ and differential probabilities of a round function of an $l$-round $\otimes$-Markov cipher, we get conditions that the cipher is $\otimes_{\mathbf W,\mathrm{ch}}$-markovian. We describe $\otimes_{\mathbf W,\mathrm{ch}}$-markovian permutations on $\mathbb Z_n$ based on an exponential operation and a logarithmic operation, which are defined on $\mathbb Z_n$ and $\mathrm{GF}(n+1)$.