On the constructions of Markov maps

Markov cipher models are used in linear analysis and differential analysis of block ciphers. We consider mappings defined on algebraic objects of various types (Abelian groups, vector spaces, etc.) and having Markovian properties. A lower estimate for the cardinality of a set $K$ of values of a random variable $k$ defining the Markov map of Abelian groups ${f_{k}}(x)$ is established.