We present new protocols for Byzantine agreement in the synchronous and authenticated setting, tolerating the optimal number of f faults among n = 2f + 1 parties. Our protocols achieve an expected O(1) round complexity and an expected O(n~2) communication complexity. The exact round complexity in expectation is 10 for a static adversary and 16 for a strongly rushing adaptive adversary. For comparison, previous protocols in the same setting require expected 29 rounds.
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