Adaptive Byzantine Agreement with O(nf) messages and O(f) rounds is possible with optimal resilience, and asynchronous agreement is shown to require Omega(n+t^2) messages.
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From Few to Many Faults: Optimal Adaptive Byzantine Agreement
Adaptive Byzantine Agreement with O(nf) messages and O(f) rounds is possible with optimal resilience, and asynchronous agreement is shown to require Omega(n+t^2) messages.