Using cloud storage services, users can store their data in the cloud to avoid the expenditure of local data storage and maintenance. To ensure the integrity of the data stored in the cloud, many data integrity auditing schemes have been proposed. In most, if not all, of the existing schemes, a user needs to employ his private key to generate the data authenticators for realizing the data integrity auditing. Thus, the user has to possess a hardware token (e.g. USB token, smart card) to store his private key and memorize a password to activate this private key. If this hardware token is lost or this password is forgotten, most of the current data integrity auditing schemes would be unable to work. In order to overcome this problem, we propose a new paradigm called data integrity auditing without private key storage and design such a scheme. In this scheme, we use biometric data (e.g. iris scan, fingerprint) as the user's fuzzy private key to avoid using the hardware token. Meanwhile, the scheme can still effectively complete the data integrity auditing.We utilize a linear sketch with coding and error correction processes to confirm the identity of the user. In addition, we design a new signature scheme which not only supports blockless verifiability, but also is compatible with the linear sketch. The security proof and the performance analysis show that our proposed scheme achieves desirable security and efficiency.