In this paper, we develop an identity-based encryption (IBE) scheme, namely MU-IBE, that achieves post-quantum security. Our scheme relies on multivariate public key cryptography, which is one of the most promising candidates of post-quantum cryptography. The proposed IBE is efficient as it incurs low communication and computation costs. Our design is proven to be IND-ID-CCA (believed to be the right security model for IBE) secure in the random oracle model under the hardness of the MQ problem. Moreover, the proposed MU-IBE is resistant to the collusion attack. In particular, our scheme is the first to achieve IND-ID-CCA in the context of multivariate identity-based encryption systems.