This paper proposes a new public key authenticated encryption (signcryption) scheme based on the hardness of q-Diffie-Hellman problems in Cap Diffie-Hellman groups, This new scheme is quite efficient: the signcryption operation has almost the same cost as an El Carnal encryption while the reverse operation only requires one pairing evaluation and three exponentiations. The scheme's chosen-ciphertext security is shown to be related to the hardness of the q-Diffie-Hellman Inversion (q-DHI) problem in the random oracle model while its unforgeability is proved under the q-Strong Diffie-Hellman assumption (q-SDH). It also provides detachable signatures that are unlinkable to the original anonymous ciphertext. We also show that most of the sender's workload can be computed offline. Our construction is based on a signature scheme independently studied by Boneh-Boyen and Zhang et al. in 2004.