Quadratic worst-case message complexity for State Machine Replication in the partial synchrony model

Abstract: We consider the message complexity of State Machine Replication protocols dealing with Byzantine failures in the partial synchrony model. A result of Dolev and Reischuk gives a quadratic lower bound for the message complexity, but it was unknown whether this lower bound is tight, with the most efficient known protocols giving worst-case message complexity $O(n^3)$. We describe a protocol which meets Dolev and Reischuk's quadratic lower bound, while also satisfying other desirable properties. To specify these properties, suppose that we have $n$ replicas, $f$ of which display Byzantine faults (with $n\geq 3f+1$). Suppose that $\Delta$ is an upper bound on message delay, i.e. if a message is sent at time $t$, then it is received by time $ \text{max} { t, GST } +\Delta $. We describe a deterministic protocol that simultaneously achieves $O(n^2)$ worst-case message complexity, optimistic responsiveness, $O(f\Delta )$ time to first confirmation after $GST$ and $O(n)$ mean message complexity.