Abstract: Using the cloud storage technology, users can outsource their data to the cloud. Such outsourcing meets the requirements of saving hardware costs and simplifying data management, because they no longer store any copies of the data in their local memory, and users cannot fully ensure whether the outsourced data are intact overall. Further, considering the client’s constrained computing power and the large size of the outsourced data, the client cannot take the extra time and effort to verify the data correctness in cloud environment. Therefore, ensuring the integrity of the outsourced data would lead to many security threats. In order to solve this problem, in this paper, we present lattice-based provable data integrity for checking the integrity of the data in the cloud. The proposed scheme not only detects any violations of client data in the cloud, but also has been proven to be safe in a random oracle. In particular, as opposed to schemes based on factoring or discrete log, the proposed scheme resists the cryptanalysis by quantum algorithms. Moreover, the proposed protocol has three other good attributes, namely support for data dynamics, computing on signed data, and multi-client verification. Finally, we present a comparison of the existing data integrity verification mechanism, as well as some open problems of lattice-based provable data integrity.