Multivariate cryptosystems are divided into public rules for which tools of encryption are open for users and systems of El Gamal type for which encryption function is not given in public and for its generation opponent has to solve discrete logarithm problem in affine Cremona group. Infinite families of transformations of free module Kⁿ over finite commutative ring K such that the degrees of their members are not growing with iteration are called stable families of transformations. Such families are needed for practical implementations of multivariate cryptosystems of El Gamal type. New explicit constructions of such families and families of stable groups and semigroups of transformations of free modules are given. New methods of creation of cryptosystems which use stable transformation groups and semigroups, and homomorphisms between them are suggested. The security of these schemes is based on a complexity of decomposition problem for element of affine Cremona semigroup into product of given generators. Proposed schemes can be used for the exchange of semigroup messages in a form of elements of a free module and for a secure delivery of multivariate maps which could be encryption tools and instruments for digital signatures.