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 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.

**Quadratic worst-case message complexity for state-machine-replication in the partial synchrony model**, pdf.

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