### Thread: Question about size of keyspace, message space, and ciphertext space

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#### Question about size of keyspace, message space, and ciphertext space

Hi everyone,

I look forward to being a contributing member of the forum (especially for anything related to mathematics). I have a question that relates to Shannon's Theorem:

Shannon's Theorem stipulates that for an encryption scheme (Gen, Enc, Dec), if we assume that |K| = |M| = |C| that the key-generation algorithm Gen must choose a secret key uniformly from the set of all possible keys, and that for every plaintext message and ciphertext there exists a single key mapping the plaintext to the ciphertext.

My question is, is it always true that if |K| = |M| then |K|=|M|=|C| or is this only when if for every plaintext message and ciphertext there exists a single, unique key that maps the plaintext to the ciphertext?

If this is not true (there is more than one key k that maps a plaintext to a ciphertext) then is it correct to stipulate that the encryption scheme would no longer be perfectly secret? My intuition is that it in fact would still be perfectly secret because the probability distributions of m in M would still be independent from the probability distributions of c in C because every plaintext m could still be mapped to any ciphertext c with the correct key. What do you guys think?

Thanks and I apologize if I'm too vague here. If you need me to clarify anything then let me know and I will.

--Decave
Last edited by Decave; September 13th, 2012 at 12:03 PM. Reason: Fixing up some awkward words/sentences