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The data security of fog computing is a key problem for the Internet of things. Identitybased encryption (IBE) from lattices is extremely suitable for fog computing. It is able to not only simplify certificate management, but also resist quantum attacks. In this paper, firstly, we construct a novel efficient lattice-based IBE scheme with Combined Public Key (CPK) technique by keeping from consumptive trapdoor generation algorithm and preimage sampling algorithm, which is required by the existing lattice-based IBE schemes based on learning with errors (LWE). In addition, its key storage cost is lower and it is IND-ID-CPA secure in the random oracle model. Furthermore, based on this, an enhanced lattice-based IBE scheme with IND-ID-CCA security is developed by employing strong one-time signature. Our schemes only need O (n3/logn) additions of vectors, while the existing schemes need at least O (n3) of additions and multiplications in Setup and Extract phase.