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An identity-based broadcast encryption cryptosystem enable senders to efficiently broadcast cipher texts to a large set of receivers, which can use arbitrary strings as public keys. In this kind of cryptosystem, if the Private Key Generator (PKG) is untrusted, you may suddenly learn that a private key corresponding to your Identity is used to decrypt some ciphertexts. What causes it happens? Except a key compromise, perhaps it must be the PKG who is acting dishonestly and trying to make malicious activities by selling your key, which is a key escrow problem. In order to solve it, we introduce the concept of Accountable Authority Identity Based Broadcast Encryption (AA-IBBE) and design a concrete scheme, which is a new approach to mitigate the key escrow problem existing by restricting the ways in which the PKG can misbehave in identity based broadcast encryption schemes. Lastly we also give its correctness proof and security analysis.