Byzantine Vector Consensus in Complete Graphs

TL;DR

This paper establishes tight bounds and provides algorithms for Byzantine vector consensus in complete graphs, addressing both synchronous and asynchronous systems with up to f Byzantine failures.

Contribution

It derives necessary and sufficient conditions for exact and approximate Byzantine vector consensus, including explicit algorithms and tight bounds.

Findings

In synchronous systems, n >= max(3f+1, (d+1)f+1) is necessary and sufficient.

In asynchronous systems, n >= (d+2)f+1 is necessary and sufficient for approximate consensus.

Constructive algorithms are provided for both exact and approximate Byzantine vector consensus.

Abstract

Consider a network of n processes each of which has a d-dimensional vector of reals as its input. Each process can communicate directly with all the processes in the system; thus the communication network is a complete graph. All the communication channels are reliable and FIFO (first-in-first-out). The problem of Byzantine vector consensus (BVC) requires agreement on a d-dimensional vector that is in the convex hull of the d-dimensional input vectors at the non-faulty processes. We obtain the following results for Byzantine vector consensus in complete graphs while tolerating up to f Byzantine failures: * We prove that in a synchronous system, n >= max(3f+1, (d+1)f+1) is necessary and sufficient for achieving Byzantine vector consensus. * In an asynchronous system, it is known that exact consensus is impossible in presence of faulty processes. For an asynchronous system, we prove that n >= (d+2)f+1 is necessary and sufficient to achieve approximate Byzantine vector consensus. Our sufficiency proofs are constructive. We show sufficiency by providing explicit algorithms that solve exact BVC in synchronous systems, and approximate BVC in asynchronous systems. We also obtain tight bounds on the number of processes for achieving BVC using algorithms that are restricted to a simpler communication pattern.