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This question is about the security of public-key ciphers when related plaintexts are encrypted by the same encryption key. Take Paillier for example. I know that Paillier is semantically secure under the DCRA. Now a sequence of related plaintexts are encrypted by Paillier with the same encryption key, but for each of the plaintexts, a fresh randomness is used in the encryption. The attacker does not know the values of the plaintexts, but he knows the relation between them. The attacker eavesdrops the ciphertexts of the sequence of plaintexts. An extreme case is that the sequence of plaintexts is the same one, in which case the attacker obtains multiple ciphertexts of a single plaintext and he knows this. Will security be reduced in this case where the attacker knows the relation between the plaintexts?

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Look at the definition of semantic security. It is for this exact reason that auxiliary information is modeled in the definition. This ensures that nothing is revealed even when the attacker knows partial information.

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