Two ID-based authenticated group key agreement schemes, proposed by Choi et al and Du et al, are insecure against an impersonation attack and they only discussed the static group. This paper proposed a variant of BD protocol, which is fully authenticated by a proven-secure ID-based signature scheme. The protocol can resist the impersonation attack, and other security attributes are also satisfied. Compared with Choi et al and Du et al schemes, the proposed one is more efficient and applicable for dynamic groups.