Secure Message Transmission on Directed Networks

Consider a sender and a receiver as two distant nodes in a network. The sender wishes to transmit a secret to the receiver, but faces an adversary controlling an unknown set of nodes. We characterize the directed networks for which there exist \epilson-secret and \epsilon-strongly secure communication protocols (for all \epsilon > 0): if all nodes are obedient the receiver learns the secret with probability at least 1−\epsilon and no information is leaked (secrecy), and this property is maintained under every strategy of the adversary (security). For secrecy, a necessary and sufficient condition is that there is a directed path from the sender to the receiver, and for each possible adversarial coalition A, there is an undirected path from the sender to the receiver that contains no node in A. For security, a necessary and sufficient condition is that for every possible adversarial coalition A, the graph obtained by removing all nodes in A still has the previous property