Group signature with message-dependent opening (GS-MDO) is a kind of group signature in which only the signers who have created group signatures on problematic messages will be identified. In the previous GS-MDO scheme, however, the number of problematic messages is bounded owing to a limitation of the Groth-Sahai proofs. In this paper, we propose the first GS-MDO scheme with the unbounded-MDO functionality in the random oracle model. Our unbounded GS-MDO scheme is based on the short group signature scheme proposed by Boneh, Boyen, and Shacham and the Boneh-Franklin identity-based encryption scheme. To combine these building blocks and to achieve CCA-anonymity, we also construct a special type of multiple encryption. This technique yields an efficient construction compared with the previous bounded GS-MDO scheme: the signature of our scheme contains about 16 group elements (3630 bits), whereas that of the previous scheme has about 450 group elements (75820 bits).