To address the issues of user privacy leakage and message authentication in Vehicular Ad hoc NETwork (VANET), an authenticatable privacy-preserving scheme based on signcryption from lattice was proposed. Firstly, the public key of receiver was used to signcrypt the message to generate the ciphertext, and only the receiver with corresponding private key could decrypt the ciphertext, which ensures messages visible only to authorized users. Secondly, after decrypting the message, the receiver calculated the hash value of the message by one-way secure hash function, and judged whether the hash value of the message changed, which realized message authentication. Finally, Number Theoretic Transform (NTT) algorithm was used to reduce the computational overhead of polynomial multiplication and improve the computational efficiency of the scheme. The proposed scheme was proved to have INDistinguishability under Chosen Ciphertext Attack (IND-CCA2) and Strong UnForgeability under Chosen Message Attack (SUF-CMA) under the random oracle model. In addition, the security of the proposed scheme is based on lattice hardness problems, so that it can resist quantum algorithm attack. Simulation experiment results show that the proposed scheme improves the performance in terms of communication delay (at least reducing 10.01%), message loss rate (at least reducing 31.79%) and communication overhead (at least reducing 31.25%) compared to similar authenticated privacy-preserving schemes and a lattice-based signature scheme. Therefore, the proposed scheme is more suitable for resource-constrained VANETs.