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May it be possible that the trusted third party knows the parties' private keys in an “ideal model” of a Secure multiparty computation protocol?

I wonder if it may be possible that a trusted third party knows the private keys (e.g. the encryption's private key or the digital signature's private key) in an "ideal model" counterpart of a Secure MPC protocol.

The trusted third party may use such knowledge to simulate authenticated (and publicly verifiable) payments from one party to another, etc...

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Yes.

Trusted party setup is just way to think about what the end goal of a particular MPC system is. So you may start with a party (lets call it the dealer) that sends it's private key to the trusted party, and then the trusted party uses it to commit a transaction.

Now replace the trusted party with a multi-party setup. The dealer here sends shares of its private key to each of the parties. The parties then can regenerate the private key or commit a transaction with it.

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