Secure message transmission on directed networks J´erˆome Renault∗, Ludovic Renou† & Tristan Tomala‡ December 7, 2010

Extended abstract Consider a sender S and a receiver R as two distant nodes in an directed graph ~ The sender has some private information (a secret), unknown to the receiver and G. all other nodes in the graph. There is a collection A of potential adversaries; each A ∈ A is a set of nodes in the graph (excluding S and R). For instance, A might be the collection of all set of nodes with at most k elements. Secret communication between the sender and the receiver is possible if there exists a protocol (profile of behavioral strategies) such that the following two requirements hold: if all nodes abide by the protocol, 1) the receiver correctly learns the secret of the sender and 2) no adversary A ∈ A gets additional information about the secret. Strongly secure communication between the sender and the receiver is possible if there exists a protocol such that the following requirements hold: 1) secret communication between the sender and the receiver is possible, 2) for any adversary A ∈ A, for any deviation of the adversary A, the receiver correctly learns the secret with arbitrary high probability and no adversary, including A, gets additional information about the secret. ∗

GREMAQ-TSE, Manufacture des Tabacs, Universit Toulouse 1, 31000 Toulouse, France.

jerome.renault @tse-fr.eu † Department of Economics, University of Leicester, Leicester LE1 7RH, United Kingdom. [email protected] ‡ Department of Economics and Finance, HEC School of Management, 78351 Jouy-en-Josas Cedex, France. [email protected]

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~ We show that secret communication is possible if and only if the the graph G is weakly A-connected, i.e., for each adversary A ∈ A, there exists a (directed or undirected) path from S to R that does not intersect A. Additionally, we show that strongly secure communication is possible if and only if ~ \ A has a strongly 1-connected subgraph that is A-connected, for each A ∈ A, the G i.e., for each A0 ∈ A, there exists a (directed or undirected) path from S to R that ~ \ A. does not intersect A0 \ A in G We relate these results to the problem of (partial) implementation of social choice functions on networks.

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