Efficient Byzantine Fault Tolerance Supplemental Material

Proof: If a correct server s executes o with sequence number i, it must have received f + 1 valid COMMIT messages for 〈o, i〉 from a quorum Q of servers. The proof is by contradiction. Suppose there is another correct server s′ that executes o with sequence number i′ > i. By the MinBFT algorithm, this can only happen if it receives f + 1 valid COMMIT messages for 〈o, i′〉, from a quorum Q′ of f + 1 servers. Since n = 2f + 1 and |Q|+|Q′| = 2f+2 ≥ 2f+1, there must be at least one server r (called intersection server) that sends COMMIT messages both for 〈o, i〉 and 〈o, i′〉. Assuming that the primary on view v is the server p, we have to consider four cases: 1) the primary and the intersection server are correct: in this case it is trivial to see that the primary will not generate two UI s for the same request operation o, so the intersection server will never send two valid COMMIT messages for 〈o, i〉 and 〈o, i′〉. 2) the primary is correct and the intersection server is faulty: The (faulty) intersection server would only be able to send valid COMMIT messages for 〈o, i〉 and 〈o, i′〉 if these messages contained UIp = 〈i,H(o)〉p and UI ′ p = 〈i′, H(o)〉p, respectively. Since the primary is