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TL;DR: It's not secure, due to MITM attacks Just because Bob receives r + 1, it doesn't mean that he is really talking to Alice. What if Mallory intercepted the key exchange of k and intercepts messages like so: Mallory intercepts Bob's key exchange to Alice. Bob and Mallory do the key exchange, Bob thinks that Mallory is Alice. Mallory and Alice do the ...

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The methods suggested in the comments are useful in practice, but they base on some specific key exchange protocols. It's useful to proove that a public key encryption scheme can be created from any such $KE^{eav}$-secure protocol. Notation Let $\langle s_A, m_A \rangle, \langle s_B, m_B \rangle$ denote the state and transmitted message from Alice and Bob ...

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Promoting my comment to an answer: Encryption hides information from someone who doesn't know the decryption key. In your case, $P_2$ knows the decryption key, and can therefore learn $x$. This is really no different than sending $x$ in the clear to $P_2$. Note that this is the strange example from the book that is secure in the malicious setting but not ...

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