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No, because then you could calculate $z_1 \oplus z_2 = (m_1 \land m_2) \oplus (m_1 \lor m_2) = m_1 \oplus m_2$. In practice, you can only find $m_1 \oplus m_2$ if both $m_1$ and $m_2$ are encrypted with the same OTP (i.e., $(m_1 \oplus y_1) \oplus (m_2 \oplus y_1) = m_1 \oplus m_2$). So without any knowledge of $m_1$, $m_2$, $y_1$ or $y_2$, there is no way ...


What you are describing is One-Time-Pad encryption, and yes it does have perfect secrecy. Note that for any ciphertext $y$ there is exactly one key $k'$ for each possible plaintext $x'$ so that $E_k(x') = y$. So if you choose the key uniformly at random the ciphertext gives no information on the plaintext, because any plaintext is equally likely.

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