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I know that you can create a ciphertext c' using c and then find the corresponding m' for c' which you can use to find m for c. So, doesn't this mean that it is not semantically secure? But I also read that it is semantically secure when it meets the DDH assumption. Is there some terminology I'm misunderstanding because that seems contradictory to me.

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Finding the corresponding $m'$ using $c'$ is a chosen cipher text attack. It's possible for a scheme to be semantically secure under certain types of attacks (perhaps something weaker like chosen-plaintext attack), but be broken under heavier attacks (like the chosen ciphertext attack you talk about).

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Oh- that helps clear up a lot of things I misinterpreted. Thanks! –  user1136342 Apr 5 '13 at 6:16
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