Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Byzantine Lattice Agreement in Synchronous Systems   In this paper, we study the Byzantine lattice agreement problem in
synchronous systems. The lattice agreement problem in crash failure model has
been studied both in synchronous and asynchronous systems, which leads to the
current best upper bound of $O(\log f)$ rounds in both systems. However, very
few algorithmic results are known for the lattice agreement problem in
Byzantine failure model. The paper [Nowak et al., DISC, 2019] first gives an
algorithm for a variant of the lattice agreement problem on cycle-free lattices
that tolerates up to $f < n/(h(X) + 1)$ Byzantine faults, where $n$ is the
number of processes and $h(X)$ is the height of the input lattice $X$. The
recent preprint by Di et al. studies this problem with a slightly modified
validity condition in asynchronous systems. They present a $O(f)$ rounds
algorithm by using the reliable broadcast primitive as a first step and
following the similar algorithmic framework as the algorithms in crash failure
model.
  In this paper, we propose three algorithms for the Byzantine lattice
agreement 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 total
number of processes. The second algorithm runs in $3\log n + 3$ rounds and
takes $O(n^2 \log n)$ messages. The third algorithm takes $4 \log f + 3$ rounds
and takes $O(n^2 \log f)$ messages. All algorithms can tolerate up to $f <
\frac{n}{3}$ Byzantine failures.

13
The categories of this paper are: quant-ph
7
The categories of this paper are: hep-th
29
The categories of this paper are: funct-an
33
The categories of this paper are: plasm-ph
25
The categories of this paper are: patt-sol
34
The categories of this paper are: chem-ph
22
The categories of this paper are: chao-dyn
32
The categories of this paper are: atom-ph
16
The categories of this paper are: stat
28
The categories of this paper are: alg-geom
15
The categories of this paper are: nucl-ex
35
The categories of this paper are: mtrl-th
21
The categories of this paper are: adap-org
14
The categories of this paper are: math-ph
20
The categories of this paper are: acc-phys
37
The categories of this paper are: supr-con