Tight Bounds for Connectivity and Set Agreement in Byzantine Synchronous Systems

TL;DR

This paper establishes tight bounds on the number of rounds needed for $k$-set agreement in Byzantine synchronous systems, showing that Byzantine failures can require one additional round compared to crash failures.

Contribution

It provides tight topological bounds on the rounds needed for $k$-set agreement under Byzantine failures, extending previous crash-failure results.

Findings

$ ext{ceil}(t/k)+1$ rounds are necessary for $k$-set agreement with Byzantine failures.

The connectivity bound is tight, with solutions existing exactly at $ ext{ceil}(t/k)+1$ rounds.

Byzantine failures may require one extra round compared to crash failures for agreement.

Abstract

In this paper, we show that the protocol complex of a Byzantine synchronous system can remain $(k - 1)$-connected for up to $\lceil t/k \rceil$ rounds, where $t$ is the maximum number of Byzantine processes, and $t \ge k \ge 1$. This topological property implies that $\lceil t/k \rceil + 1$ rounds are necessary to solve $k$-set agreement in Byzantine synchronous systems, compared to $\lfloor t/k \rfloor + 1$ rounds in synchronous crash-failure systems. We also show that our connectivity bound is tight as we indicate solutions to Byzantine $k$-set agreement in exactly $\lceil t/k \rceil + 1$ synchronous rounds, at least when $n$ is suitably large compared to $t$. In conclusion, we see how Byzantine failures can potentially require one extra round to solve $k$-set agreement, and, for $n$ suitably large compared to $t$, at most that.