# plaintext distribution ,perfect secrecy cryptosystem

I have been trying to solve the next problem:
let pi be a perfect secrecy cryptosystem with specific plaintext probability distribution. How to prove pi will have perfect secrecy for all other possible plaintext probability distributions?

• I'm not sure that the claim actually holds. Are you sure that there are no additional constraints on the "specific plaintext distribution"? Unless I'm confused, any encryption scheme (or even the identityfunction) offers perfect secrecy for the distribution that has probability 0 for all but one plaintext. But obviously this does not imply perfect secrecy for other distributions. – Maeher Nov 16 '14 at 19:22

Perfect scheme \Pi defined such that for every distribution over M and every $c\in C$ it holds that: $Pr[C=c|M=m]=Pr[C=c]$ which means that probability to get a cipher text $c$ is independent of the message been encrypted, thus independent of the distribution over the message space M.
• However, "a perfect secrecy cryptosystem with specific plaintext probability distribution" would obviously be defined differently. $\;$ – user991 Nov 17 '14 at 23:49