Assume $|K| = |P| = |C| = 1$ with $K$ being the set of all keys, $P$ the set of all messages and $C$ the set of all cypher texts. So for one message $P_1$, there exists exactly one key $K_1$ such that $E_{K_1}(P_1) = C_1$. It can be easily calculated that this encryption is perfectly secure, so $I(P,C) = 0.$ But this doesn't make any sense to me. An encryption like this is anything else but "secure" since there is only one key.
2026-02-22 19:49:48.1771789788
Encryption with $|K| = |P| = |C| = 1$ is perfectly secure?
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An eavesdropper has exactly the same information about the plaintext after observing the ciphertext as before seeing it. (there is only one message, after all). So absurd though it may seem, it fulfills the definition for perfectly secure.
Slightly more realistic: a one bit message, a one bit key (chosen completely random) and xor as the operation. This is just a slength one one-time pad and is also perfectly secure.