Tight Bounds for Connectivity and Set Agreement in Byzantine Synchronous Systems

In this paper, we show that the protocol complex of a Byzantine synchronous system can remain (k− 1)-connected for up to ⌈t/k⌉ rounds, where t is the maximum number of Byzantine processes, and t ≥ k ≥ 1. Protocol complex connectivity is important since a (k − 1)-connected protocol complex does not admit solutions for the k-set agreement problem. In crash-failure systems, the connectivity upper bound is ⌊t/k⌋ rounds, therefore, in Byzantine systems, the ambiguity in the communication can potentially persist for one extra round, delaying the solution to k-set agreement and other related problems. Additionally, we show that our connectivity bound is tight, at least when n+1, the number of processes, is suitably large compared to t. We solve a formulation of k-set agreement appropriate for Byzantine systems in ⌈t/k⌉ + 1 rounds. Essentially, we see that Byzantine failures can potentially require us one extra round to handle ambiguity, and, in specific cases, at most that.