ON THE KEY EXCHANGE WITH MATRICES OF LARGE ORDER AND GRAPH BASED NONLINEAR MAPS

Abstract

In the paper we discuss the group theoretical algorithm of Diffie - Hellman key exchange in the cases of symmetrical group $S_{p^n}$ and more general Cremona group of polynomial automorphisms of free module ${\mathbb{K}}^n$ over arbitrary commutative ring $\mathbb{K}$. We show that conjugation of affine map with nonlinear polynomial map $f$ can be element of large order and small degree. Same properties hold for each element of cyclic group generated by such elements. We consider some algorithms for generation of subgroups of large order and small degree of their elements.