Byzantine agreement with homonyms

Abstract

So far, the distributed computing community has either assumed that all the processes of a distributed system have distinct identifiers or, more rarely, that the processes are anonymous and have no identifiers. These are two extremes of the same general model: namely, $n$ processes use $\ell$ different identifiers, where $1 \le \ell \le n$. In this paper, we ask how many identifiers are actually needed to reach agreement in a distributed system with $t$ Byzantine processes. We show that having $3t+1$ identifiers is necessary and sufficient for agreement in the synchronous case but, more surprisingly, the number of identifiers must be greater than $\frac{n+3t}{2}$ in the partially synchronous case. This demonstrates two differences from the classical model (which has $\ell =n$): there are situations where relaxing synchrony to partial synchrony renders agreement impossible; and, in the partially synchronous case, increasing the number of correct processes can actually make it harder to reach agreement. The impossibility proofs use the fact that a Byzantine process can send multiple messages to the same recipient in a round. We show that removing this ability makes agreement easier: then, $t+1$ identifiers are sufficient for agreement, even in the partially synchronous model, assuming processes can count the number of messages with the same identifier they receive in a round.