On the key exchange and multivariate encryption with nonlinear polynomial maps of stable degree

We say that the sequence g_n, n≥3, n→∞ of polynomial transformation bijective mapsof free module Kgⁿ over commutative ring K is a sequence of stable degree if the order of g_n is growing with n and the degree of each nonidentical polynomial map of kind g^k_n is an independent constant c. Transformation b = τg_n^kτ−1, where τ is the affine bijection, n is large and k is relatively small, can be used as a base of group theoretical Diffie-Hellman key exchange algorithm for the Cremona group C(Kⁿ) of all regular automorphisms of Kⁿ. The specific feature of this method is that the order of the base may be unknown for the adversary because of the complexity of its computation. The exchange can be implemented by tools of Computer Algebra (symbolic computations). The adversary can not use the degree of right handside in b^x = d to evaluate unknown x in this form for the discrete logarithm problem. In the paper we introduce the explicit constructions of sequences of elements of stable degree for the cases c = 3 and c = n+2/4 for each commutative ring K containing at least 3 regular elements and discuss the implementation of related key exchange and multivariate map algorithms.