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Families of stable cyclic groups of nonlinear polynomial transformations of affine spaces Kn over general commutative ring K of increasing with n order can be used in the key exchange protocols and related to them El Gamal multivariate cryptosystems. To use high degree of noncommutativity of affine Cremona group correspondents have to modify multivariate El Gamal algorithm via the usage of conjugations for two polynomials of kind gk and g−1 given by key holder (Alice) or giving them as elements of different transformation groups. The idea of hidden tame homomorphism and comlexity of decomposition of polynomial transwormation into word of elements of Cremona semigroup can be used. We suggest usage of new explicit constructions of infinite families of large stable subsemigroups of affine Cremona group of bounded degree as instruments of multivariate key exchange protocols. Recent results on generation of families of stable transformations of small degree and density via technique of symbolic walks on algebraic graphs are observed. Some of them used for the implementation of schemes as above with feasible computational complexity. We consider an example of a new implemented quadratic multivariate cryptosystem based on the above mentioned ideas.

On desynchronised multivariate algorithms of El Gamal type for stable semigroups of affine Cremona group