On the key exchange with matrices of large order and graph based nonlinear maps

In the paper we discuss the group theoretical algorithm of Die -Hellman key exchange in the cases of symmetrical group Spn and more generalCremona group of polynomial automorphisms of free module Kn over arbitrarycommutative ring K. We show that conjugation of ane map with nonlinearpolynomial map f can be element of large order and small degree. Sameproperties hold for each element of cyclic group generated by such elements.We consider some algorithms for generation of subgroups of large order andsmall degree of their elements.