An IND-CPA secure encryption scheme might not conceal identities, in the following sense: given a pair of ciphertexts C, C′ for equal-length messages, it might be “obvious” if the ciphertexts were encrypted using the same random key or were encrypted using two different random keys. Give an example of a (plausibly) IND-CPA secure encryption scheme that has this is identity revealing. Then give a definition for “identity-concealing” encryption. Your definition should imply IND-CPA security but a scheme meeting your definition can’t be identity-revealing.
How can a scheme that is identity revealing be IND-CPA? I understand that the view of changing the key is not the same as changing the message but the point of ensuring security is to make sure that no additional information is revealed.