Byzantine Lattice Agreement in Synchronous Message Passing Systems

We propose three algorithms for the Byzantine lattice agreement problem in synchronous systems. The first algorithm runs in min{3h(X) + 6, 6 √ fa + 6}) rounds and takes O(n2 min{h(X), √ fa}) messages, where h(X) is the height of the input lattice X, n is the total number of processes in the system, f is the maximum number of Byzantine processes such that n ≥ 3f + 1 and fa ≤ f is the actual number of Byzantine processes in an execution. The second algorithm takes 3 log n + 3 rounds and O(n2 log n) messages. The third algorithm takes 4 log f + 3 rounds and O(n2 log f) messages. All algorithms can tolerate f < n3 Byzantine failures. This is the first work for the Byzantine lattice agreement problem in synchronous systems which achieves logarithmic rounds. In our algorithms, we apply a slightly modified version of the Gradecast algorithm given by Feldman et al [10] as a building block. If we use the Gradecast algorithm for authenticated setting given by Katz et al [12], we obtain algorithms for the Byzantine lattice agreement problem in authenticated settings and tolerate f < n2 failures. 2012 ACM Subject Classification Theory of computation → Distributed algorithms