Proxy re-encryption scheme based on BB2 identity based encryption

In 1998, Blaze, Bleumer, and Strauss proposed a kind of cryptographic primitive called proxy re-encryption. In proxy re-encryption, a proxy can transform a ciphertext computed under Alice's public key into one that can be opened under Bob's decryption key. They predicated that proxy re-encryption and re-signature will play an important role in our life. In 2007, Matsuo proposed the concept of four types of re-encryption schemes: CBE to IBE(type 1), IBE to IBE(type 2), IBE to CBE (type 3), CBE to CBE (type 4 ). Now CBE to IBE and IBE to IBE proxy re-encryption schemes are being standardized by IEEEP1363.3 working group. One feature of proxy re-encryption schemes proposed by Matsuo is that they are all based on BB1 identity based encryption.We ask question like this: can we construct proxy re-encryption schemes based on BB2 identity based encryption? We give affirmative answer to this question. We construct an IBE to IBE proxy re-encryption scheme based on BB2 with the help of PKG and prove its security in the standard model.