Optimal algorithms for synchronous Byzantine -set agreement.

This article is on k -set agreement ( k SA) in an n -process synchronous message-passing system in which up to t processes can commit Byzantine failures. k SA 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 k SA in the presence of Byzantine processes. To this end it first extends a previous result and shows that k SA cannot be solved when ( n < 2 t + t k ) ∧ ( n − t ≥ k + 1 ). On the positive side, it presents two synchronous round-based algorithms that solve Byzantine k SA. 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 > 2 t + 1 and solves k SA for k ≥ ⌊ n − t n − 2 t ⌋ + 1, while the second assumes n ≤ 2 t + 1 and solves k SA 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 = 3 t and solves 2SA. The second assumes 2 t + 1 < n ≤ 3 t and solves k SA where k = n − t n − 2 t is an integer. It is worth noticing that the results presented in this article “almost” complete the possibility/impossibility cartography of k SA in synchronous Byzantine systems. The only case that remains open is when n − t n − 2 t is not an integer. • Optimal algorithms for k-set agreement in synchronous systems where some processes are Byzantine • Statement of non-trivial impossibility results • Knowledge-based algorithms • Introduction of new proof-oriented concepts (Squares and Regions) • Round Optimal algorithms (1 & 2 rounds)