  At Eurocrypt'99, Paillier presented a public-key cryptosystem based on a
novel computational problem. It has interested many researchers because it was
additively homomorphic. 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 transformed into a signature scheme but these
variants miss the feature. In particular, we simplify the alternative
decryption procedure of Bresson-Catalano-Pointcheval 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.
