Crash | Omission | Byzantine | |
---|---|---|---|

Synchrony | $f<n$ is possible $f+1$ round executions must exist |
$f \geq n/2$ is impossible | $f<n/2$ possible with PKI / PoW $f \geq n/3$ impossible without PKI/PoW |

Partial Synchrony | $f \geq n/2$ is impossible | $f<n/2$ is possible | $f<n/3$ is possible $f \geq n/3$ is impossible |

Asynchrony | non terminating executions must exist | $f<n/2$ possible in $O(1)$ expected | $f<n/3$ possible in $O(1)$ expected |

Here $n$ is the number of parties, and $f$ is the number of parties that the adversary controlls. Recall that **Synchrony** $\subseteq$ **Partial Synchrony** $\subseteq$ **Asynchrony**. Similarly that **Crash** $\subseteq$ **Omission** $\subseteq$ **Byzantine**. Therefore,

- Any upper bound holds if you go up and/or to the left in the table. e.g., the $O(1)$ expected round upper bounds under asynchrony also hold in partial synchrony and in synchrony.
- Any lower bound holds if you go down and/or to the right in the table. e.g., the impossibility of $f \geq n/3$ with Byzantine adversaries in partial synchrony carries over to asynchrony, and the $t+1$ round lower bound carries over from crash to omission and Byzantine.

Acknowledgments: many thanks to Kartik Nayak for help with this post!

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