Byzantine-Resilient Population Protocols

Population protocols model information spreading in networks where pairwise node exchanges are determined by an external random scheduler. Most of the population protocols in the literature assume that the participating $n$ nodes are honest. Such assumption may not be, however, accurate for large-scale systems of small devices. Hence, in this work, we study a variant of population protocols, 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 adversary that has complete knowledge of all the node states to the Weak 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 $O(\log n)$ states per node. We also give an alternative algorithm with $d = \Omega(\min\{f \log^2 n + 1,n\})$. Moreover, we combine both algorithms into one using random coins. The only other known previous work on Byzantine-resilient population protocols tolerates up to $f = o(\sqrt n)$ faulty nodes and works against a static adversary; hence, our protocols significantly improve the tolerance by an $\omega(\sqrt n)$ factor and all of them work correctly against a stronger dynamic adversary.