Byzantine Geoconsensus

TL;DR

This paper introduces the geoconsensus problem for processes in a plane with Byzantine fault areas, proving impossibility results and providing algorithms that tolerate a significant number of Byzantine processes under geometric constraints.

Contribution

It defines the geoconsensus problem, establishes impossibility results, and presents new algorithms for Byzantine fault tolerance based on geometric coverage and process distribution.

Findings

Impossibility of geoconsensus with up to three fault areas.

Algorithms tolerating up to approximately half of processes Byzantine under geometric conditions.

Extensions to higher dimensions and various fault area shapes.

Abstract

We define and investigate the consensus problem for a set of $N$ processes embedded on the $d$-dimensional plane, $d\geq 2$, which we call the {\em geoconsensus} problem. The processes have unique coordinates and can communicate with each other through oral messages. In contrast to the literature where processes are individually considered Byzantine, it is considered that all processes covered by a finite-size convex fault area $F$ are Byzantine and there may be one or more processes in a fault area. Similarly as in the literature where correct processes do not know which processes are Byzantine, it is assumed that the fault area location is not known to the correct processes. We prove that the geoconsensus is impossible if all processes may be covered by at most three areas where one is a fault area. Considering the 2-dimensional embedding, on the constructive side, for $M \geq 1$ fault areas $F$ of arbitrary shape with diameter $D$, we present a consensus algorithm that tolerates $f\leq N-(2M+1)$ Byzantine processes provided that there are $9M+3$ processes with pairwise distance between them greater than $D$. For square $F$ with side $\ell$, we provide a consensus algorithm that lifts this pairwise distance requirement and tolerates $f\leq N-15M$ Byzantine processes given that all processes are covered by at least $22M$ axis aligned squares of the same size as $F$. For a circular $F$ of diameter $\ell$, this algorithm tolerates $f\leq N-57M$ Byzantine processes if all processes are covered by at least $85M$ circles. We then extend these results to various size combinations of fault and non-fault areas as well as $d$-dimensional process embeddings, $d\geq 3$.