Based on the discrete logarithm, this paper proposes a group signature schemes. The scheme preserves the main merits inherent in previously proposed schemes, and solves the anonymity problem. In the scheme, even though the group authority announces some additional information to provide the verifier with confirmation of the identity of the signer, the anonymity of this signer will be maintained in the other group signatures, and the group authority need not renew any of the keys of this signer. The reason is that announced information is only provided in order to identify the specific group signature.