Certificateless encryption scheme with non-black-box technology

Certificateless public key cryptography is intermediate between identity-based and traditional PKI-based cryptography. Since the introduction of the notion of certificateless cryptography, it has got extensively improved. But it has been proved that it was impossible for a protocol to be secure against both the strong Type II adversary and malicious KGC adversary in the Al-Riyami and Paterson´s model. This paper presents a different security model with non-black-box technology and gives an efficient protocol secure against both the strong Type I adversary and malicious KGC adversary in the non-black-box model. Furthermore, the efficient security model indicates that non-black-box reduction model has the more advantage than the black-box reduction model again.