# Perfect secrecy and change of plaintext probability distributions

How to prove that if a cryptosystem has perfect secrecy for a given plaintext probability distribution then it will have perfect secrecy for all other possible plaintext probability distributions?

Do we have to make any additional assumptions on probability distributions (probability of each plaintext is not zero?) to prove this?

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Welcome to Crypto.SE, Student! Can you show us what you've tried so far? Where did you get stuck? Have you tried proving it for some special cases, like specific examples of a non-uniform probability distribution on the plaintext? Can you give us an example of a probability distribution where you are unable to prove your desired result? Make sure you read the FAQ and our advice on asking these sorts of questions. – D.W. Apr 16 '13 at 1:22
Another way to approach the problem is to attempt to find a counterexample; can you devise a (possibly toy) system where probability distribution A has perfect secrecy, but probability distribution B doesn't. Finding such an example gives you criteria that need to hold for such a theorem to work; how attempts weren't counterexamples may give you a clue as to why such a theorem was true (and hence point you to a proof) – poncho Apr 16 '13 at 14:48