A note on certificate-based encryption

Identity-based encryption (IBE), introduced by Shamir, has a major issue, which is that the private key generator (PKG) can decrypt any ciphertext with the master secret key. This problem is called the inherent key escrow in IBE schemes. To solve this problem, Gentry proposed certificate-based encryption. In this paper, from the idea of Gentry we propose two IBE schemes, which can avoid key escrow problem. Asymmetric bilinear pairings on ordinary elliptic curves are of use and identities is mapped to integers. The first scheme is a generalization from Gentry´s scheme. Our scheme saves one pairing in encryption algorithm. Our second scheme is an IBE scheme with attributes of fine-grained access control, which is based on the cipher-policy attribute-based encryption (ABE) of Water. This schemes combines properties of IBE and ABE schemes. It would be interesting to solve the key escrow problem for ABE schemes.