Optimal algorithms for synchronous Byzantine k-set agreement

This article is on k-set agreement (kSA) in an n-process synchronous message-passing system in which up to t processes can commit Byzantine failures. kSA is a decision problem in which at least each correct (i.e., non-Byzantine) process is assumed to propose and decide a value such that at most k different values are decided by the correct processes, in such a way that, if all the correct processes propose the same value v, they will decide v. This article investigates the possibility/impossibility domain of kSA in the presence of Byzantine processes. To this end it first extends a previous result and shows that kSA cannot be solved when (n < 2t + t/k) boolean AND (n - t >= k + 1). On the positive side, it presents two synchronous round-based algorithms that solve Byzantine kSA. These algorithms are optimal with respect to the value of k and the number of rounds. The first algorithm is a one-round algorithm that has two instances. The first assumes n > 2t + 1 and solves kSA for k >= [ n-t n-2t ] + 1, while the second assumes n <= 2t + 1 and solves kSA for k >= t +1 (so this second case does not require a majority of correct processes). The second algorithm is based on two new notions denoted Square and Regions that allow each correct process to locally build a global knowledge on which processes proposed which values. This algorithm has also two instances. The first assumes n = 3t and solves 2SA. The second assumes 2t + 1 < n <= 3t and solves kSA where k = n-t n-2t is an integer. It is worth noticing that the results presented in this article "almost" complete the possibility/impossibility cartography of kSA in synchronous Byzantine systems. The only case that remains open is when (n-t)/n-2t is not an integer.