⊗W, _ch-markovian transformations

Let X be an alphabet of plaintexts (ciphertexts) of iterated block ciphers and (X, ®) be a regular abelian group. The group operation ® defines the difference of a text pair. ®-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 ®-Markov ciphers with independent round keys, the sequence of round differences forms a Markov chain. In this paper, we consider ®-Markov ciphers and a partition W = {W ₀,..., W _r-1} with blocks being lumped states of the Markov chain. An /-round ®-Markov cipher is called ® _W;ch-markovian if the cipher and W satisfy the following condition: the block numbers sequence j ₀,..., j such that, for all i е {0,...,/}, the i -round difference belongs to is a Markov chain. This definition can be also extended for permutations on X. For a partition W and differential probabilities of a round function of an /-round ®-Markov cipher, we get conditions that the cipher is ® _W;ch-markovian. We describe ® _W;ch-markovian permutations on Z _n based on an exponential operation and a logarithmic operation, which are defined on Z _n and GF(n +1).

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