Topology-Hiding Computation on all Graphs

Abstract

A distributed computation in which nodes are connected by a partial communication graph is called topology-hiding if it does not reveal information about the graph beyond what is revealed by the output of the function. Previous results have shown that topology-hiding computation protocols exist for graphs of constant degree and logarithmic diameter in the number of nodes [Moran-Orlov-Richelson, TCC\u2715; Hirt \etal, Crypto\u2716] as well as for other graph families, such as cycles, trees, and low circumference graphs [Akavia-Moran, Eurocrypt\u2717], but the feasibility question for general graphs was open. In this work we positively resolve the above open problem: we prove that topology-hiding computation is feasible for all graphs under either the Decisional Diffie-Hellman or Quadratic-Residuosity assumption. Our techniques employ random-walks to generate paths covering the graph, upon which we apply the Akavia-Moran topology-hiding broadcast for chain-graphs (paths). To prevent topology information revealed by the random-walk, we design multiple random-walks that, together, are locally identical to receiving at each round a message from each neighbors and sending back processed messages in a randomly permuted order