# Distribution of Conditional Probability

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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I assume $C$ is the ciphertext, $K$ is the key, $P$ is the plaintext, and $e_k(x)$ is the encrypted text corresponding to plaintext $x$ and key $k$.
$\text{Pr}(C=y|P=x)$ is not a random variable, so it doesn't quite make sense to say that is uniform. If you mean, for a given plaintext $x$ that $C$ is uniform (over the set of all possible ciphertexts), that's usually not true because the space of possible keys is usually much smaller than the space of possible ciphertexts.