Population protocols model information spreading and computation in network systems where pairwise node exchanges are determined by an external random scheduler and nodes have small memory. Most of the population protocols in the literature assume that the participating $n$ nodes are honest. Such an assumption may not be, however, accurate for large-scale systems of small devices. Hence, in this work, we study population protocols in a setting where up to $f$ nodes can be Byzantine. We examine the majority (binary) consensus problem against different levels of adversary strengths, ranging from the Full Byzantine adversary that has complete knowledge of all the node states to the Weak Content-Oblivious Byzantine adversary that has only knowledge about which exchanges take place. We also take into account Dynamic vs Static node corruption by the adversary. We give lower bounds that require any algorithm solving the majority consensus to have initial difference $d = \Omega(f + 1)$ for the tally between the two proposed values, which holds for both the Full Static and Weak Dynamic adversaries. We then present an algorithm that solves the majority consensus problem and tolerates $f \leq n / c$ Byzantine nodes, for some constant $c>0$, with $d = \Omega(f + \sqrt{n \log n})$ and $O(\log^3 n)$ parallel time steps, using $O(\log^3 n)$ states per node. We also give an alternative algorithm, with the same asymptotic performance, for $d = \Omega(\min\{f \log^2 n + 1,n\})$. Finally, we combine both algorithms into one using a new robust distributed common coin. The only other known previous work on Byzantine-resilient population protocols tolerates up to $f = o(\sqrt n)$ faulty nodes and was analyzed against a Static adversary; hence, our protocols significantly improve fault-tolerance by an $\omega(\sqrt n)$ factor and all of them work correctly against a stronger Dynamic adversary.

翻译：种群协议模拟了网络系统中的信息传播和计算，其中成对节点交换由外部随机调度器决定，且节点具有较小的内存。文献中的大多数种群协议假设参与的 $n$ 个节点是诚实的。然而，这种假设对于大规模小型设备系统可能并不准确。因此，在本工作中，我们在最多 $f$ 个节点可能为拜占庭节点的设置下研究种群协议。我们针对不同程度的敌方强度研究多数（二进制）共识问题，从完全了解所有节点状态的完全拜占庭敌手，到仅了解哪些交换发生的弱内容无关拜占庭敌手。我们还考虑了敌方对节点的动态腐败与静态腐败。我们给出了下界，要求任何解决多数共识问题的算法在两个提议值的计票之间具有初始差异 $d = \Omega(f + 1)$，该下界同时适用于完全静态和弱动态敌手。接着，我们提出了一种算法，该算法解决多数共识问题，并能容忍 $f \leq n / c$ 个拜占庭节点（其中 $c>0$ 为某个常数），具有 $d = \Omega(f + \sqrt{n \log n})$ 和 $O(\log^3 n)$ 并行时间步，每个节点使用 $O(\log^3 n)$ 个状态。我们还给出了一种替代算法，具有相同的渐近性能，适用于 $d = \Omega(\min\{f \log^2 n + 1, n\})$。最后，我们通过一种新的鲁棒分布式公共硬币将两种算法合并为一个算法。先前唯一已知的关于拜占庭容错种群协议的工作仅能容忍最多 $f = o(\sqrt n)$ 个故障节点，并且是在静态敌手环境下分析的；因此，我们的协议在容错性上实现了 $\omega(\sqrt n)$ 倍的显著提升，并且所有协议都能在更强的动态敌手下正确工作。