With the rise of digital transformation, cloud storage has become a mainstream solution for managing large-scale data due to its scalability and cost-efficiency. However, outsourcing data to the cloud deprives users of physical control, introducing significant risks of data tampering or loss, which in turn motivates the development of cloud storage auditing. Due to the advantages of certificate-based cryptography, many certificate-based cloud storage data integrity auditing (CB-CSDIA) protocols have been proposed in recent years. Nevertheless, existing CB-CSDIA protocols suffer from limitations in terms of security or efficiency. In this paper, we analyze the core techniques and underlying issues of current CB-CSDIA constructions. To provide a rigorous theoretical foundation for CB-CSDIA, we formally define certificate-based quasi-linearly homomorphic signatures (CB-QLHS) and establish a comprehensive security model. A generic construction from CB-QLHS to CB-CSDIA is then proposed. Furthermore, we present an efficient instantiation of CB-QLHS and prove its security in the random oracle model. Performance evaluations demonstrate that auditing protocol built upon our CB-QLHS construction outperform existing solutions.