In 2008, Guo and Zhang  introduced an accountable authority IBBE scheme, which was a combination of Goyal's  first scheme and Delerablee's  IBBE scheme.
We construct a universal scheme for realizing AB-IBBE, such that any IND-ID-CPA secure IBBE scheme can be extended to an IND-ID-CPA secure AB-IBBE scheme.
Using our universal AB-IBBE scheme, any IND-ID-CPA secure IBBE scheme can be converted into an AB-IBBE scheme.
Following , we define the security model for IBBE which is equivalent to the full security in IBE schemes.
We call an adversary A in the above game a IND-ID-CCA (chosen ciphertext security for IBBE systems under chosen identity attacks) adversary.
We say that if the above indistinguishability game allow no decryption oracle query, then the IBBE scheme is only chosen plaintext (IND-ID-CPA) secure.
To the best of our knowledge, our IBBE is the first efficient scheme that is full security with constant size private key and ciphertext.