Group encryption (GE) is the encryption analogue of group signatures. It allows a sender to verifiably encrypt a message for some certied but anonymous member of a group. The sender is further able to convince a verifier that the ciphertext is a well-formed encryption under some group member's public key. As in group signatures, an opening authority is empowered with the capability of identifying the receiver if the need arises. One application of such a scheme is secure repository at an unknown but authorized cloud server, where the archive is made accessible by a judge order in the case of misbehavior, like a server hosting illegal transaction records (this is done in order to balance individual rights and society's safety). In this work we describe Traceable GE system, a group encryption with refined tracing capabilities akin to those of the primitive of raceable signatures" (thus, balancing better privacy vs. safety). Our primitive enjoys the properties of group encryption, and, in addition, it allows the opening authority to reveal a user-specific trapdoor which makes it possible to publicly trace all the ciphertexts encrypted for that user without harming the anonymity of other ciphertexts. In addition, group members are able to non-interactively prove that specific ciphertexts are intended for them or not. This work provides rigorous definitions, concrete constructions in the standard model, and security proofs.