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The traditional MPC definition of correctness has no notion of correctness on the inputs. The traditional MPC correctness property deals with the output, i.e., the protocol is correct if $y$ where $y=f(x_1,x_2,\dots,x_n)$ is guaranteed to be output. What the $x_1,\dots,x_n$ values are is completely up to the inputting party. So, if you want to check that ...


If the honest parties have commitments to the malicious parties' inputs or [encapsulations generated by honest parties] of those inputs, then the function can be modified to check those. ‚Äč Otherwise, the value computed by the trusted party "taking honest party inputs and modified inputs of corrupted parties" by definition results in a correct output.


As the previous answer says, they are certainly NOT the same. However, there is certainly a connection between them. Specifically, the covert model just says that there is a deterrent parameter $\epsilon$ and the guarantee is that if the adversary tries to cheat then it will be caught with probability at least $\epsilon$. The question that arises is how ...

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