At Eurocrypt'99, Paillier presented a public-key cryptosystem based on a novel computational problem. It has interested many researchers because of its additively homomorphic property. In this paper, we show that there is a big difference between the original Paillier's encryption and some variants. The Paillier's encryption can be naturally converted into a signature scheme but these variants miss the feature. In particular, we simplify the alternative decryption procedure of

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... ointcheval encryption scheme proposed at Asiacrypt'03. The new version is more applicable to cloud computing because of its double trapdoor decryption mechanism and its flexibility to be integrated into other cryptographic schemes. It captures a new feature that its two groups of secret keys can be distributed to different users so as to enhance the robustness of key management.