Iterative approximate Byzantine consensus in arbitrary directed graphs

This paper identifies necessary and sufficient conditions for the existence of iterative algorithms that achieve approximate Byzantine consensus in arbitrary directed graphs, where each directed link represents a communication channel between a pair of nodes. The class of iterative algorithms considered in this paper ensures that, after each iteration of the algorithm, the state of each fault-free node remains in the convex hull of the states of the fault-free nodes at the end of the previous iteration. We present the necessary and sufficient condition for the existence of such iterative consensus algorithms in synchronous arbitrary point-to-point networks in presence of Byzantine faults in two different equivalent forms. We prove the necessity using an indistinguishability argument. For sufficiency, we develop a proof framework, which first uses a series of “transition matrices” to model the state evolution of the fault-free nodes using our algorithm, and then proves the correctness by identifying important properties of the matrices. The proof framework is useful for other iterative fault-tolerant algorithms. We discuss the extensions to asynchronous systems and the Byzantine links fault model.