Internet of Things (IoT) devices upload their data into the cloud for storage because of their limited resources. However, cloud storage data has been subject to potential integrity threats, and consequently auditing techniques are demanded to ensure the integrity of stored data. Unfortunately, existing auditing approaches require owners to undertake expensive tag calculations, which is unsuitable for resource-constrained IoT devices. To resolve the issue, we present a F air C loud A uditing proposal by employing the B lockchain (FCAB). We combine certificateless signatures with the designed dynamic structure to constructively offload the cost of tag computation from the IoT device to the introduced fog node, significantly reducing the local burden. Considering that fog nodes may behave dishonestly during auditing, FCAB enables the IoT device to verify the audit result's authenticity by extracting reliable checking records from the blockchain, thereby achieving auditing fairness, which ensures that the honest cloud and fog node will gain the corresponding reward. Finally, FCAB is proved to satisfy tag unforgeability, proof unforgeability, privacy preserving, and auditing fairness. Experiment evaluations affirm that FCAB is computationally and communicationally efficient and retains a smaller and fixed computation locally at the data processing stage (mainly including tag computation) than existing auditing methods.