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In Cryptography, the conditional probability..

$$\text{Pr} (C = y \mid P = x) = \sum_{k,\,e_k(x)=y} \text{Pr}(K = k)$$

If the keys are uniformly distributed, can we say..

$\mathrm{Pr}(C=y\mid P=x)$ is also uniform i.e ciphertext is uniformly distributed too. If yes, can someone give me how to mathematically show it?

Also, can we say that every ciphertext uses a different key, since ever key is equally likely?

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