A Diophantine model of routes in structured P2P overlays

An important problem in any structured Peer-to-Peer (P2P) overlay is what routes are available between peers. Understanding the structure of routes helps to solve challenging problems related to routing performance, security, and scalability. In this paper, we propose a theoretical approach for describing routes. It is based on a recent result in the linear Diophantine analysis and introduces a novel Diophantine model of P2P routes. Such a route aggregates several P2P paths that packets follow. A commutative context-free grammar describes the forwarding behavior of P2P nodes. Derivations in the grammar correspond to P2P routes. Initial and final strings of a derivation define packet sources and destinations, respectively. Based on that we construct a linear Diophantine equation system, where any solution counts forwarding actions in a route representing certain integral properties. Therefore, P2P paths and their composition into routes are described by a linear Diophantine systems; its solutions (basis) define a structure of P2P paths.

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