Strong law of large numbers on graphs and groups

Abstract

Abstract. We consider (graph-)group-valued random element ξ, discuss the properties of a mean-set E(ξ), and prove the generalization of the strong law of large numbers for graphs and groups. Furthermore, we prove an analogue of the classical Chebyshev's inequality for ξ and Chernoff-like asymptotic bounds. In addition, we prove several results about configurations of mean-sets in graphs and discuss computational problems together with methods of computing mean-sets in practice and propose an algorithm for such computation.

Mean-set attack: cryptanalysis of Sibert et al. authentication protocol

“…It is proved in [27] that for any distribution µ on V (Γ) either domain(M ) = ∅ or domain(M ) = V (Γ). In the case when domain(M ) = V (Γ), we say that M (·) is totally defined.…”

“…Using the Cayley graph construction one can similarly define a notion of the mean-set for a finitely generated group G (relative to a fixed generating set). Similar mean values (in different settings) are used rather often; see [27] for some history and literature sources. Below, we recall some results proved in [27].…”

“…Similar mean values (in different settings) are used rather often; see [27] for some history and literature sources. Below, we recall some results proved in [27].…”

“…Sibert et al authentication protocol, is an example of an interactive (dynamic, randomized) proof system. In this paper, we use probabilistic tools, introduced in [27] and outlined in Section 2.3 below, to design an attack on this particular cryptographic primitive and show that it is not computationally zero-knowledge. In addition, we conduct some experiments that support our conclusions and show that the protocol is not secure in practice.…”

“…It turns out that the same is true in general infinite groups. One can generalize a number of mathematical tools of the classical probability theory to finitely generated groups (see [27] and Section 2.3 below) in order to have the counterparts of (AV1), (AV2), and (AV3). Indeed,…”