A Diophantine Model of Routes in Structured P2P Overlays

Abstract — An important intrinsic property of any structured Peer-to-Peer (P2P) overlay is multi-hop paths. Understanding their structure helps to solve challenging problems related to routing performance, security, and scalability. In this paper, we introduce a mathematical Diophantine model of P2P routes. Such a route aggregates several P2P paths that messages follow. A commutative context-free grammar describes forwarding behavior of P2P nodes. Derivations in the grammar correspond to P2P routes. Initial and final strings of a derivation define message 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 system; its solutions (basis) define a structure of P2P paths. I