Secure message transmission on directed networks

A sender wishes to transmit a secret to a receiver through a communication network, where some nodes are controlled by an adversary. We characterize the directed networks for which there exist ε -secret and ε -strongly secure communication protocols (∀ε>0∀ε>0): if all nodes are obedient the receiver learns the secret with probability at least 1−ε1−ε 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.