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This work was done by Laasya Bangalore, Ashish Choudhury, and Arpita Patra, and appears at PODC’18, the ACM Symposium on Principles of Distributed Computing 2018.

The problem of Byzantine Agreement (BA) is of interest to both distributed computing and cryptography community. Following well-known results from the distributed computing literature, BA in the asynchronous network setting encounters inevitable non-termination issues.

The impasse is overcome via randomization that allows construction of BA protocols in two flavours of termination guarantee – with overwhelming probability and with probability one. The latter type termed as almost-surely terminating BAs are the focus of this paper. An eluding problem in the domain of almost-surely terminating BAs is achieving a constant expected running time. Our work makes progress in this direction. In a setting with n parties and an adversary with unbounded computing power controlling atmost t parties (who can deviate from the protocol in any manner), we present two almost-surely terminating BA protocols in the asynchronous setting. Our first protocol runs in expected time linear in n and tolerates optimal corrupt parties i.e. t < n/3. Existing protocols in this setting either run for expected quadratic (in n) time or exponential computing power from the honest parties. In terms of communication complexity, our construction outperforms all known constructions that offer almost-surely terminating feature. Our Second protocol runs in constant expected running time (independent of n) and tolerates slightly less than one-third corrupt parties i.e. t < n/(3 + ε) for some constant ε > 0. The known constructions with constant expected running time either require ε ≥ 1 implying t < n/4 or calls for exponential computing power from the honest parties.