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# 33% Impossibility Result

Last updated Jul 3, 2022 Edit Source

In general (assuming a partially-synchronous system model), a protocol can achieve safety all the time and additionally liveness in synchronous conditions if and only if $n \geq 3f + 1$ (equivalently, $f < \frac n 3$)

This result holds despite the assumption of PKI (unlike the PSL-FLM result), so this bound must be driven by the possibility of unbounded message delays.

Intuition:

1. We can’t wait indefinitely for all nodes to respond (one valid strategy for Byzantine nodes is to never respond, even after GST) so realistically we can only wait to hear from $n - f$ nodes before deciding on the next possible action
2. But we can’t say for certain the $f$ nodes are actually Byzantine (they could be honest nodes that are congested), thus of the $n - f$ nodes, more than half should be honest, $f < \frac 1 2 (n -f)$ or equivalently $f < \frac n 3$