A supposedly easy problem in information theory . . .

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Chu
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Here is the problem I am having trouble with:

Prove that for any cipher that has perfect secrecy, the size of the key space is at least as large as the size of the cyphertext space.

For those rusy with information security, this essentially means proving that for each message p in Plaintext space P, and each encrypted message c in Cyphertext space C :

P[p|c] = P[p]
----------------

For some reason I am pretty much completly stuck. From the birthday problem it is trivial to show that more then one mapping from p->c uses the same key, and if the probabilities of choosing a specific key is different then choosing any other key, the problem is pretty trivial.

The case where all the keys can be chosen with the same probability (for example |K| = 1/2 |C| = 1/2 |P| ) is driving me nuts. Can someone give me some nudge in the right direction on solving this problem?
 

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