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I've been given as an exercise to show an encryption scheme that is CCA-secure but is not a secure message transmission scheme.

What is a message transmission scheme, and how does it differ from a regular encryption scheme?

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An "encryption scheme" defines the encryption/decryption of data.

A "message transmission scheme" is about securing transmission and defines both "privacy" and "authenticity" between a sender and a receiver.

Since you haven't asked about the definition of CCA-secure (encryption) schemes and since you've been given this as an exercise, I won't mention anything about CCA-secure schemes because that would quickly provide an answer which would almost solve your exercise for you (and you wouldn't learn from that). But, when you keep in mind what I wrote above, you should be able to quickly come up with an encryption scheme that is CCA-secure while not being a secure transmission scheme.

Hint:

Your CCA-secure scheme will be missing something a "message transmission scheme" provides… look closely at what I wrote, ask yourself what a "CCA-secure scheme" defines at a minimum, and then start thinking. You should be able to wrap your brain around it easily if you look at what each scheme defines.

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    $\begingroup$ Isn't it true that any CCA-secure encryption scheme that doesn't provide authenticity embedded in it will not be a secure message transmission scheme? $\endgroup$
    – Bush
    Commented Dec 7, 2013 at 16:55
  • $\begingroup$ @Bush Correct, CCA security does not imply authenticated encryption. Since you're on the correct path, I can provide an example: encrypting a single block with a block cipher is supposed to be CCA-secure (except for the fact that it isn't randomized — yet, you could add a nonce to a short message and then encrypt) but it doesn't offer any authentication. Note that this is a very simplified (meaning: not copy-and-paste ready) example. But I think you can take it from here and come up with a better example, can't you? ;) $\endgroup$
    – e-sushi
    Commented Dec 7, 2013 at 17:26

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