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In the scenario where an entity has control of greater than 51% of the hash power and wishes to build a longer chain where they double-spend previously confirmed transactions, do they broadcast the chain to the network as they are working on it?

The alternative scenario I see is the entity working on the chain and only broadcasting it to full nodes at the point where it has overtaken what is believed to be the current longest chain. This second scenario would appear to me to be far more damaging to confidence in Bitcoin if it was pulled off. Technically, are both these scenarios possible?

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Yes, both scenarios are possible. The validity of a block is determined solely by its contents, not by when it is broadcast or received, so the attacker's longer chain is equally valid whether they broadcast it as they go, or all at once.

(There is an exception to this principle on testnet because of the 20-minute rule, but that's not relevant here.)

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