Remote data integrity checking (RDIC) enables a data storage server, say a cloud server, to prove to a verifier that it is actually storing a data owner's data honestly. To date, a number of RDIC protocols have been proposed in the literature, but most of the constructions suffer from the issue of a complex key management, that is, they rely on the expensive public key infrastructure (PKI), which might hinder the deployment of RDIC in practice. In this paper, we propose a new construction of identity-based (ID-based) RDIC protocol by making use of key-homomorphic cryptographic primitive to reduce the system complexity and the cost for establishing and managing the public key authentication framework in PKI-based RDIC schemes. We formalize ID-based RDIC and its security model, including security against a malicious cloud server and zero knowledge privacy against a third party verifier. The proposed ID-based RDIC protocol leaks no information of the stored data to the verifier during the RDIC process. The new construction is proven secure against the malicious server in the generic group model and achieves zero knowledge privacy against a verifier. Extensive security analysis and implementation results demonstrate that the proposed protocol is provably secure and practical in the real-world applications.