# Perfect Secrecy, Encryption

An encryption scheme $(\mathrm{Gen},\mathrm{Enc},\mathrm{Dec})$ over a message space $M$ is perfectly secret if and only if for every probability distribution over $M$, every message $m\in M$, and every ciphertext $c\in C$, $$\mathrm{Pr}[C=c\mid M=m]=\mathrm{Pr}[C=c].$$

Please can you help me prove this?

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Hint: What is your definition of "perfectly secret"? What this says is that knowing the message $m$ gives you no information about the cyphertext. A perfect encryption scheme says that knowing the encrypted text gives you no information about the message. These look like inverses.