Early stopping in Byzantine agreement

Abstract

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.

Uniform consensus is harder than consensus

“…The lower bound presented here is very close to the one established by Dolev, Reischuk, and Strong [9]: they prove that consensus requires at least min(f + 2, t + 1) rounds before all correct processes can halt, i.e., cease executing the algorithm. As already pointed out in [9], it is important to notice the difference between the time at which a process can decide and the time at which it can halt.…”

“…In such early deciding algorithms for consensus, processes decide one round earlier than in any uniform consensus algorithm for most cases (0 f t − 2). By refining time complexity analysis as in [9], we thus show that uniform consensus is harder than consensus for the crash failure model.…”

“…Moreover, Merritt [27] showed that t + 1 is a lower bound on the number of rounds required for deciding in the worst case for both of these problems (see Chapter 6 in [26] and Section 3 infra for more detailed references concerning this result). Following [9], we refine this analysis by discriminating runs according to the number of failures f that actually occur. We prove that uniform consensus requires at least f + 2 rounds whereas consensus requires only f + 1 rounds if f is less than t − 1, and both consensus and uniform consensus only require f + 1 rounds if f = t − 1 or f = t.…”

“…As already pointed out in [9], it is important to notice the difference between the time at which a process can decide and the time at which it can halt. From the worst case lower bound, we prove that consensus requires f + 1 rounds to decide; hence by the lower bound in [9], correct processes cannot stop just after making a decision, in an early deciding algorithm. In turn, our lower bound result implies that, in any early stopping algorithm, processes must postpone deciding to the very end of the computation in order to guarantee agreement uniformity.…”

Efficient Secure Multi-party Computation

“…There exist also various techniques which improve the efficiency of (stand-alone) protocols for Byzantine agreement, e.g. "early stopping" [DRS82]. However, they lead to "staggered termination", and it is unclear how and whether at all they are applicable for multi-party computation protocols.…”