Let K be a general finite commutative ring. We refer to a family gn, n = 1, 2, . . . of bijective polynomial multivariate maps of Knas a family with invertible decomposition gn= gn1gn2. . . g gnk, such that the knowledge of the composition of gniallows computation of gnifor O(ns) (s > 0) elementary steps. A polynomial map g is stable if all non-identical elements of kind gt, 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.