The emergence of the vehicular ad-hoc network (VANETs) could meet the need for an efficient and safe transportation. Vehicles wirelessly interact with nearby road-side units (RSUs) and send the data to the cloud servers (CS), but this approach has limitations (i.e., network congestion and privacy leakage). Edge computing can effectively solve problems of network congestion, feedback delay and privacy leakage during transmission caused by CS. Thus, this paper proposes a certificateless aggregate signcryption with online/offline encryption (CLOASC) based on edge computing (EDC). Correspondingly, certificateless signcryption (CLSC), online/offline encryption (OOE) and aggregate signcryption technology solve the key escrow problem, shorten the encryption time and realize the quick verification of multiple users, respectively. Furthermore, it is worth mentioning that the scheme we propose does not have any bilinear pairings or scalar multiplications in the online encryption stage, which greatly reduces the computational cost in the online stage. In the random oracle model, our scheme was proved to be existential unforgeability under adaptive chosen message attacks (EUF-ACM) and adaptive indistinguishability under chosen-ciphertext attack (IND-CCA2) security. Simulation results show that our scheme is computationally and communication based lightweight as compared to the related.