On new symbolic key exchange protocols and cryptosystems based on a hidden tame homomorphism - Ustimenko, V.A. (Reports of the National Academy of Sciences of Ukraine, 2018)
Multivariate cryptosystems are divided into public rules, for which tools of encryption are open for users and systems of the El Gamal type, for which the encryption function is not given in public, and, for its generation, the opponent has to solve a discrete logarithm problem in the affine Cremona group. Infinite families of transformations of a free module Kn over a 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 the 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 the decomposition problem for an element of the affine Cremona semigroup into a product of given generators. Proposed schemes can be used for the exchange of 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.