Anonymous ABE is a promising primitive for enforcing fine-grained access control for the big data as well as preserving privacy of the users in Cloud Computing. However, traditional anonymous ABE schemes may not be secure in the real word due to the side- channel attacks. In addition, the existing anonymous ABE schemes are considered in the leak-free scenario assuming that secret keys are not leaked to the adversary. Thus, it is compelling to study the anonymity of ABE schemes in the context of key leakage attacks. Aiming at tackling the challenge above, an anonymous CP-ABE scheme against side- channel attacks in the bounded-leakage model is constructed. As a main technique tool, the dual system encryption technique is adopted. The proposed scheme uses LSSS as access structures and achieves adaptive security in the standard model. In addition, the results in simulation experiments indicate that the proposed scheme is efficient and practical.