Optimal Algorithms for Synchronous Byzantine k-Set Agreement

Abstract

Considering a system made up of n processes prone to Byzantine failures, k-set agreement allows each process to propose a value and decide a value such that at most k different values are decided by the correct (i.e., non-Byzantine) processes, in such a way that, if all the correct processes propose the same value v, they will decide v (when $k=1$, k-set agreement boils down to consensus). This paper presents a two-round algorithm that solves Byzantine k-set agreement on top of a synchronous message-passing system. This algorithm is based on two new notions denoted by Square and Regions which allow processes to locally build a global knowledge on which processes proposed some values. Two instances of the algorithm are presented. Assuming $n=3t$, where t is the maximum number of Byzantine, the first instance solves 2-set agreement. The second one solves the more general case $2t < n \le 3t$, where $k=\frac{n-t}{n-2t}$ is an integer. These two algorithm instances are optimal with respect to the number of rounds executed by the processes (namely two rounds). Combined with previous results, this article “nearly closes” the solvability of Byzantine k-set agreement in synchronous message-passing systems (more precisely, the only remaining case for which it is not known whether k-set agreement can or cannot be solved is when $k=\frac{n-t}{n-2t}$ is not an integer).

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