Abstract: Membership revocation is an important issue of group signatures. Verifier-local revocation （VLR for short） and witness-based method are two current main nontrivial approaches. The latter is more suitable in some environments, especially in mobile environments. In CCS04, an efficient VLR group signature scheme is proposed by Boneh and Shacham, which is very short but not having the propriety of backward unlinkability （BU）, a property that keeps the anonymity of a member even after heshe has been revoked. Recently, BU-VLR group signature schemes have been proposed. However, these schemes are not comparable with Boneh-Shachams in performance. Based on the weak DTDH and q-SDH assumptions, a new BU-VLR group signature scheme is proposed. The advantage of the proposal is shorter signature length and lower computation cost over the previous BU-VLR group signature schemes. To reduce the data published in the revocation list, an extended version is also provided with lower overhead in computation according to the method put forward by Nakanishi and Funabiki in ASIACRYPT05. Finally, an improved version of the scheme in IEICE07 proposed by Nakanish and Funabiki is given using the construction trick in signature, the signature length of which is only 77.8% that of the original scheme under the same computation.