On New Examples of Families of Multivariate Stable Maps and their Cryptographical Applications

Let K be a general finite commutative ring. We refer to a family g_n, n = 1, 2, . . . of bijective polynomial multivariate maps of K_n as a family with invertible decomposition g_n = g_n¹ g_n² . . . g g_n^k, such that the knowledge of the composition of g_nⁱ allows computation of g_nⁱ for O(n^s) (s > 0) elementary steps. A polynomial map g is stable if all non-identical elements of kind g^t, t > 0 are of the same degree. We construct a new family of stable elements with invertible decomposition. This is the first construction of the family of maps based on walks on the bipartite algebraic graphs defined over K, which are not edge transitive. We describe the application of the above mentioned construction for the development of stream ciphers, public key algorithms and key exchange protocols. The absence of edge transitive group essentially complicates cryptanalysis.