A Necessary Condition for Byzantine $k$-Set Agreement

TL;DR

This paper establishes a simple, necessary condition for solving Byzantine $k$-set agreement in various systems, proving that certain bounds on system size and resilience are fundamental and tight for specific cases.

Contribution

It introduces a new, straightforward proof of a necessary condition for Byzantine $k$-set agreement, extending understanding of system limitations under Byzantine faults.

Findings

Proves that $k$-set agreement cannot be solved if $n \, \leq \, 2t + \frac{t}{k}$ in Byzantine systems.

Provides a tight bound for Byzantine consensus ($k=1$) in synchronous message-passing systems.

Offers a simple proof technique applicable to both message-passing and shared memory systems.

Abstract

This short paper presents a necessary condition for Byzantine $k$-set agreement in (synchronous or asynchronous) message-passing systems and asynchronous shared memory systems where the processes communicate through atomic single-writer multi-reader registers. It gives a proof, which is particularly simple, that $k$-set agreement cannot be solved $t$-resiliently in an $n$-process system when $n \leq 2t + \frac{t}{k}$. This bound is tight for the case $k=1$ (Byzantine consensus) in synchronous message-passing systems.