Strong law of large numbers on graphs and groups

We introduce the notion of the mean-set (expectation) of a graph(group-) valued random element ξ and prove a generalization of the strong law of large numbers on graphs and groups. Furthermore, we prove an analogue of the classical Chebyshev’s inequality for ξ. We show that our generalized law of large numbers, as a new theoretical tool, provides a framework for practical applications; namely, it has implications for cryptanalysis of groupbased authentication protocols. In addition, we prove several results about configurations of mean-sets in graphs and their applications. In particular, we discuss computational problems and methods of computing of mean-sets in practice and propose an algorithm for such computation.