I am studying modular approach to the design and analysis of authentication and of the key exchange protocols. One of them are proposed by bellare et al. On page number 11, I do not understand the proof of theorem 3 which states that

Theorem 3: Let $\lambda$ be an MT-authenticator (i.e. $\lambda$ emulates MT in unauthenticated networks), and let $C_{\lambda}$ be a compiler constructed based on $\lambda$ as described above. Then $C_{\lambda}$ is an authenticator.

The proof uses the $A_{\pi}$ adversary in authenticated model and $U_{\pi^{'}}$ adversary in unauthenticated model.

  1. I do not understand how these two adversaries interact using $\lambda$ protocol and $C_{\lambda}$ compiler?
  2. How the whole interaction of the protocol is done under the simulated environment?

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