Privacy protection is an important security issue in today's big data information era. As one of theoretical and technical bases, cryptography can be utilized to protect several kinds of privacy information, such as content and identity. Identity-Based hash proof system is a basic cryptographic primitive, which can be used to construct lots of schemes for privacy protection. Through analyzing all existing identity-based hash proof systems based on lattices, this work reveals that one of their common deficiencies is the large bit size of ciphertext, which further results in the low efficiency of the related cryptographic schemes. Thus it is of great significance to reduce the size of their cipheretexts. In this paper, a new hash proof system is first presented based on the learning with errors assumption in the standard model, and the smoothness of the system is proved through employing the properties of discrete Gaussian distribution and smooth parameter over lattices. Then, in order to transform this new hash proof system into the identity setting, the preimage sampling function proposed by Gentry, et al. is used to sample the identity secret key for any identity id with the help of random oracle. As an extension for this new hash proof system based on lattices, an updatable hash proof system can also be obtained in the standard model. Finally, the efficiency of these new constructions is analyzed, and a comparison with other existing constructions is performed.