Accountable authority identity-based encryption (A-IBE), as an attractive way to guarantee the user privacy security, enables a malicious private key generator (PKG) to be traced if it generates and re-distributes a user private key. Particularly, an A-IBE scheme achieves full black-box security if it can further trace a decoder box and is secure against a malicious PKG who can access the user decryption results. In PKC'11, Sahai and Seyalioglu presented a generic construction for full black-box A-IBE from a primitive called dummy identity-based encryption, which is a hybrid between IBE and attribute-based encryption (ABE). However, as the complexity of ABE, their construction is inefficient and the size of private keys and ciphertexts in their instantiation is linear in the length of user identity. In this paper, we present a new efficient generic construction for full black-box A-IBE from a new primitive called token-based identity-based encryption (TB-IBE), without using ABE. We first formalize the definition and security model for TB-IBE. Subsequently, we show that a TB-IBE scheme satisfying some properties can be converted to a full black-box A-IBE scheme, which is as efficient as the underlying TB-IBE scheme in terms of computational complexity and parameter sizes. Finally, we give an instantiation with the computational complexity as O(1) and the constant size master key pair, private keys, and ciphertexts.