# Perfect secrecy of a crypto system

Suppose we have the following crypto system: $P = C = K = \{0, 1, . . . , n − 1\}$, $E_k(x) = (x + k) \bmod n$ and $D_k(y) = (y − k) \bmod n$. Prove that the crytosystem has perfect secrecy. Perfect secrecy means that the ciphertext does not leak any information about the plaintext (i.e., $P(X = x) = P(X = x \mid Y = y)$).

• I don't really see a question. Just a command. I.e., "prove this for me" instead of "how would I prove this". That said, I believe the latter has been explored on here at least once. – mikeazo Dec 17 '14 at 19:17

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.