The scheme introduced by Hwang and Liu in 2008 is shown to be insecure against the key replacement attack. A more practical attack model is introduced, and according to this model, an efficient certificateless encryption scheme is presented which is IND-CCA secure against both Type Ⅰ and Type Ⅱ attackers without random oracle. This encryption scheme can defend against the malicious key generation center attack, and can reach Girault’s trusted level 3 as in traditional public key infrastructure-based cryptography. This certificateless encryption scheme has a shorter public key than other schemes in the standard model.

This paper shows that the protocol presented by Goyal et al. can be further simplified for a one-way function, with the simplified protocol being more practical for the decisional Diffie-Hellman assumption. Goyal et al. provided a general transformation from any honest verifier statistical zero-knowledge argument to a concurrent statistical zero-knowledge argument. Their transformation relies only on the existence of one-way functions. For the simplified transformation, the witness indistinguishable proof of knowledge protocols in “parallel” not only plays the role of preamble but also removes some computational zero-knowledge proofs, which Goyal et al. used to prove the existence of the valid openings to the commitments. Therefore, although some computational zero-knowledge proofs are replaced with a weaker notion, the witness indistinguishable protocol, the proof of soundness can still go through.