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Two different kinds of Byzantine Agreement for distributed systems with processor faults are defined and compared. The first is required when coordinated actions may be performed by each participant at different times. This kind of agreement is called Eventual Byzantine Agreement (EBA). The second is needed for coordinated actions that must be performed by all participants at the same time. This kind is called Simultaneous Byzantine Agreement (SBA).This paper deals with the number of rounds of message exchange required to reach Byzantine Agreement of either kind (BA). If an algorithm allows its participants to reach Byzantine agreement in every execution in which at most t participants are faulty, then the algorithm is said to tolerate t faults. It is well known that any BA algorithm that tolerates t faults (with t < n -1 where n denotes the total number of processors) must run at least t + 1 rounds in some execution. However, it might be supposed that in executions where the number f of actual faults is small compared to t, the number of rounds could be correspondingly small. A corollary of our first result states that (when t < n -1) any algorithm for SBA must run t + 1 rounds in some execution where there are no faults. For EBA (with t < n -1), a lower bound of min(t + 1, f + 2) rounds is proved. Finally, an algorithm for EBA is presented that achieves the lower bound, provided that t is on the order of the square root of the total number of processors.

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Reaching agreement in a distributed system while handling malfunctioning behavior is a central issue for reliable computer systems. All previous algorithms for reaching the agreement required an exponential number of messages to be sent, with or without authentication.We give polynomial algorithms for reaching (Byzantine) agreement, both with and without the use of authentication protocols. We also prove that no matter what kind of information is exchanged, there is no way to reach agreement with fewer than t+l rounds of exchange, where t is the upper bound on the number of faults.