Exact Byzantine Consensus in Directed Graphs

For synchronous point-to-point n-node networks of undirected links, it has been previously shown that, to achieve consensus in presence of up to f Byzantine faults, the following two conditions are together necessary and sufficient: (i) n >= 3f + 1 and (ii) network connectivity greater than 2f. The first condition, that is, n >= 3f + 1, is known to be necessary for directed graphs as well. On the other hand, the second condition on connectivity is not necessary for directed graphs. So far, tight necessary and sufficient condition for Byzantine consensus in directed graphs has not been developed. This paper presents tight necessary and sufficient condition for achieving Byzantine consensus in synchronous networks that can be represented as directed graphs. We provide a constructive proof of sufficiency by presenting a new Byzantine consensus algorithm for directed graphs. Further work is needed to improve the message overhead of Byzantine consensus in directed graphs.