Although the existing group signature schemes on lattice can effectively resist the attacks of quantum computing, it is difficult to avoid the complicated management problem of user’s public key certificate. Based on techniques such as rejection sampling and lattice basis delegation, this paper combines the identity-based encryption with the group signature on lattice to construct an identity-based group signature on lattice in the random oracle model. First of all, the system master key is obtained from the trapdoor generation algorithm; Then, the lattice delegation technology extracts the user’s identity information and obtains the user’s private key. Finally, the signature is generated by using the rejection sampling algorithm instead of the zero-knowledge proof system in the signing stage. Meanwhile, this paper uses the LPR encryption algorithm proposed to ensure that the signature can be opened for group administrator by the traceability key. Security analysis shows that the full anonymity, unforgeability and full traceability of the proposed scheme in this paper can be reduced to the hardness assumptions of RSIS and RLWE. Compared with other group signatures on lattice, the proposed scheme is based on identity-based encryption and has certain advantages in storage overhead. Specifically, the overhead of key and signature are decreased roughly by 79.6%, 39.9%, respectively.