Byzantine Gathering in Polynomial Time

TL;DR

This paper presents a new deterministic algorithm for Byzantine gathering in networks that reduces time complexity by assuming a strong team of agents with a quadratic number of good agents relative to Byzantine ones, using small global knowledge.

Contribution

It introduces a Byzantine gathering solution with lower complexity under the assumption of a strong team and small global knowledge, improving over previous exponential-time algorithms.

Findings

Deterministic gathering achieved with quadratic team size assumption

Reduced time complexity compared to prior exponential algorithms

Effective use of small global knowledge for coordination

Abstract

We study the task of Byzantine gathering in a network modeled as a graph. Despite the presence of Byzantine agents, all the other (good) agents, starting from possibly different nodes and applying the same deterministic algorithm, have to meet at the same node in finite time and stop moving. An adversary chooses the initial nodes of the agents and assigns a different label to each of them. The agents move in synchronous rounds and communicate with each other only when located at the same node. Within the team, f of the agents are Byzantine. A Byzantine agent acts in an unpredictable way: in particular it may forge the label of another agent or create a completely new one. Besides its label, which corresponds to a local knowledge, an agent is assigned some global knowledge GK that is common to all agents. In literature, the Byzantine gathering problem has been analyzed in arbitrary n-node graphs by considering the scenario when GK=(n,f) and the scenario when GK=f. In the first (resp. second) scenario, it has been shown that the minimum number of good agents guaranteeing deterministic gathering of all of them is f+1 (resp. f+2). For both these scenarios, all the existing deterministic algorithms, whether or not they are optimal in terms of required number of good agents, have a time complexity that is exponential in n and L, where L is the largest label belonging to a good agent. In this paper, we seek to design a deterministic solution for Byzantine gathering that makes a concession on the proportion of Byzantine agents within the team, but that offers a significantly lower complexity. We also seek to use a global knowledge whose the length of the binary representation is small. Assuming that the agents are in a strong team i.e., a team in which the number of good agents is at least some prescribed value that is quadratic in f, we give positive and negative results.