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~1 message/block \cite{nakamoto}

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papers/CasperTFG/CasperTFG.tex
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@ -80,7 +80,7 @@ Traditional consensus protocols (such as multi-Paxos and pbft) are notoriously d
In the context of state machine replication, traditional protocols decide (with irrevocable finality) on one ``block'' of state transitions/transactions to add to the shared operation log at a time. To decide on a block, a node must receive $\mathcal{O}(N)$ messages, where $N$ is the number of consensus-forming nodes. In the context of state machine replication, traditional protocols decide (with irrevocable finality) on one ``block'' of state transitions/transactions to add to the shared operation log at a time. To decide on a block, a node must receive $\mathcal{O}(N)$ messages, where $N$ is the number of consensus-forming nodes.
Blockchain consensus protocols like Bitcoin do not finalize/decide on one block at a time. In fact, the Bitcoin blockchain in particular does not make ``finalized decisions'' at all; blocks are ``orphaned'' if/when they are not in the highest total difficulty chain. However, if the miners are able to mine on the same blockchain, then the blocks that get deep enough into the blockchain won't be reverted (``orphaned''). A block's depth in the blockchain therefore serves as a proxy for finalization. In the average case for blockchain consensus protocols, each node only requires approximately one message, $\mathcal{O}(1)$, for every block. Blockchain consensus protocols like Bitcoin do not finalize/decide on one block at a time. In fact, the Bitcoin blockchain in particular does not make ``finalized decisions'' at all; blocks are ``orphaned'' if/when they are not in the highest total difficulty chain. However, if the miners are able to mine on the same blockchain, then the blocks that get deep enough into the blockchain won't be reverted (``orphaned''). A block's depth in the blockchain therefore serves as a proxy for finalization. In the average case for blockchain consensus protocols, each node only requires approximately one message, $\mathcal{O}(1)$, for every block.\footnote{For Bitcoin, refer to \cite{nakamoto}, section 5 on pp. 3–4.}
Traditional consensus protocol research has focused on producing protocols that are asynchronously safe (i.e.\ blocks won't be reverted due to arbitrary timing of future events) and live in asynchrony (or partial synchrony) (i.e.\ nodes eventually decide on new blocks). On the other hand, the Bitcoin blockchain is not safe in an asynchonous network but is safe and live (for unknown block-depth or ``confirmation count'') in a ``partially synchronous network.'' Traditional consensus protocol research has focused on producing protocols that are asynchronously safe (i.e.\ blocks won't be reverted due to arbitrary timing of future events) and live in asynchrony (or partial synchrony) (i.e.\ nodes eventually decide on new blocks). On the other hand, the Bitcoin blockchain is not safe in an asynchonous network but is safe and live (for unknown block-depth or ``confirmation count'') in a ``partially synchronous network.''