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Let's focus on the UC proof in the existence of the dummy adversary $\mathcal{D}$ and the simulator $\mathcal{S}$ that internally runs $\mathcal{D}$. My questions are:

  1. Is it true that $\mathcal{S}$ can only see that the environment sends input to some real world corrupted party via $\mathcal{D}$ while $\mathcal{S}$ has no idea about the content of the input? Otherwise there is no need for $\mathcal{S}$ to extract the input of corrupted parties.
  2. Assume that there are two corrupted parties communicating with each other via authenticated channel in real world protocol. Is it true that $\mathcal{S}$ can obtain their messages from the authenticated channel (or, from $\mathcal{D}$)? Does $\mathcal{D}$ relay messages between these two corrupted parties as the real adversary in protocol execution?
  3. Can $\mathcal{S}$ see the message (including its header and content) sent from some corrupted main party to the environment in real protocol execution?

Many thanks.

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