Secure Message Transmission in Asynchronous Directed Graphs

We study the problem of secure message transmission (SMT) in asynchronous directed graphs, where an unbounded Byzantine adversary can corrupt some subset of nodes specified via an adversary structure. We focus on the particular variant (0, delta)-SMT, where the message remains perfectly private, but there is a small chance that the receiver R may not obtain it. This variant can be of two kinds: Monte Carlo where R may output an incorrect message with small probability; and Las Vegas - where R never outputs an incorrect message. For a Monte Carlo (0, delta)-SMT protocol to exist in an asynchronous directed graph, we show that the minimum connectivity required in the network does not decrease even when privacy of the message being transmitted is not required. In the case of Las Vegas (0,delta)-SMT, we show that the minimum connectivity required matches exactly with the minimum connectivity requirements of the zero-error variant of SMT - (0, 0)-SMT. For a network that meets the minimum connectivity requirements, we provide a protocol efficient in the size of the graph and the adversary structure. We also provide a protocol efficient in the size of the graph for an important family of graphs, when the adversary structure is threshold.