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In this paper, we study the Byzantine lattice agreement problem insynchronous systems. The lattice agreement problem in crash failure model hasbeen studied both in synchronous and asynchronous systems, which leads to thecurrent best upper bound of $O(\log f)$ rounds in both systems. However, veryfew algorithmic results are known for the lattice agreement problem inByzantine failure model. The paper [Nowak et al., DISC, 2019] first gives analgorithm for a variant of the lattice agreement problem on cycle-free latticesthat tolerates up to $f < n/(h(X) + 1)$ Byzantine faults, where $n$ is thenumber of processes and $h(X)$ is the height of the input lattice $X$. Therecent preprint by Di et al. studies this problem with a slightly modifiedvalidity condition in asynchronous systems. They present a $O(f)$ roundsalgorithm by using the reliable broadcast primitive as a first step andfollowing the similar algorithmic framework as the algorithms in crash failuremodel.  In this paper, we propose three algorithms for the Byzantine latticeagreement problem in synchronous systems. The first algorithm takes $\min\{3h(X) + 6,6\sqrt{f} + 6\})$ rounds and $O(n^2 \min\{h(X), \sqrt{f}\})$messages, where $h(X)$ is the height of the input lattice $X$, $n$ is the totalnumber of processes. The second algorithm runs in $3\log n + 3$ rounds andtakes $O(n^2 \log n)$ messages. The third algorithm takes $4 \log f + 3$ roundsand takes $O(n^2 \log f)$ messages. All algorithms can tolerate up to $f <\frac{n}{3}$ Byzantine failures.

Byzantine Lattice Agreement in Synchronous Systems