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My question is about symmetric randomized encryption. Let $c = \mathsf{E}(k,m;r)$ be a randomized block encryption function where $r$ denotes a random input chosen for each message block $m$. $k$ and $c$ denote the symmetric key and ciphertext respectively. If $\mathsf{E}$ has indistinguishability of $m_0$, $m_1$ for any chosen plaintexts by an adversary given a ciphertext $c$ of $m_b$ for a randomly chosen bit $b$, how does this translate to the security of key $k$ given $c_i$ for any chosen plaintexts $m_i$ by allowing oracle access to the adversary? In the deterministic encryption there is the security condition that the computation of $k$ given $m,c$ is infeasible. How is this property assured by indistinguishability?

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If you could guess the key, you could decrypt the challenge ciphertexts and figure out which plaintext is which.

Conversely, if the cryptosystem has ciphertext indistinguishability in this attack model, then you obviously can't figure out which plaintext is which, so you can't guess the key.

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