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.

Byzantine Agreement has become increasingly important in establishing distributed properties when there may exist errors in the systems. Recent polynomial algorithms for reaching Byzantine Agreement provide us with feasible solutions for obtaining coordination and synchronization in distributed systems. In this paper we study the amount of information exchange necessary to ensure Byzantine Agreement. This is measured by the number of messages and the number of signatures appended to messages (in case of authenticated algorithms) the participating processors need to send, in the worse case, in order to reach Byzantine Agreement. The lower bound for the number of signatures in the authenticated case is ~(nt), where n is the number of participating processors and t is the upper bound on the number of faults. If n is large compared to t, it matches the upper bounds from previously known algorithms. The lower bound for the number of messages is ~(n+t2). We present an algorithm that achieves this bound and for which the number of phases does not exceed the minimum t+l by more than a constant factor.

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ABS TRACT :We prove the following lower bounds for on line computation. i) Simulating two tape nondeterministic machines by one tape machines requires ~(n log log n) time.2) Simulating k tape (deterministic) machines by machines with k pushdown i/(k+ 1 ) stores requires ~(n log n) time.