Group signature schemes permit a group member signing on messages anonymously and unlinkably on behalf of a group. The anonymity can be revoked when arguments occurred. This paper proposes an efficient group signature scheme based on the discrete logarithm problem, which means no trapdoor functions are used in the settings of cryptographic parameters. Both the group public key and signature size are fixed length. The signing procedure/verifying procedure requires only 9/5 modular multi-exponentiations to generate/verify a signature. It is more efficient than previous schemes based on the same cryptographic assumption.