Byzantine-Resilient Population Protocols

TL;DR

The paper tackles binary majority consensus in population protocols under Byzantine faults, introducing a formal model with dynamic and observant adversaries on a complete graph powered by a randomized scheduler. It establishes lower bounds on the initial tally difference $d$ required to ensure correct majority and then designs three protocols—Asymmetric-C-Partial-D, Symmetric-C-Full-D, and Combined-C-D—that tolerate up to $f \le n/c_f$ Byzantine nodes while stabilizing in $O(\log^3 n)$ parallel steps using $O(\log^3 n)$ states per node. A robust distributed common coin enables combining the two main strategies into Combined-C-D, allowing the approach to adapt to unknown $f$ and still achieve correct majority under dynamic adversaries. Compared to prior work, which handled only $o(\sqrt n)$ faults, the proposed schemes offer a significant fault-tolerance boost (up to a linear fraction of $n$) with polylogarithmic running time and modest per-node memory, even against strong adversaries. The results contribute new lower bounds, phase-based designs with cancellation/duplication techniques, and a practical mechanism to blend multiple protocols for resilience in Byzantine-populated environments.

Abstract

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 $d = Ω(f + 1)$ 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 $d = Ω(f + \sqrt{n \log n})$ 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 $d = Ω(\min\{f \log^2 n + 1,n\})$. 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 $ω(\sqrt n)$ factor and all of them work correctly against a stronger Dynamic adversary.

Theorems & Definitions (39)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4: Chernoff bound