Industrial Internet of things (IIoT) devices encounter problems such as data authentication and privacy protection when collecting and storing data through the cloud. Attribute-based signature (ABS) can not only realize the data authentication, but also protect the identity privacy of the signer. In the existing server-aided ABS (SA-ABS) schemes, the computational overhead of the signer and the verifier is reduced with the help of the server, and the security of the server-aided verification phase is guaranteed by the defense of collusion attack of the signer and the server. However, none of the existing SV-ABS schemes can verify the validity of partial signature generated by the server, which will lead to a potential risk of partial signature forgery by the server. To overcome this challenge, a novel server-aided and verifiable ABS (SA-VABS) scheme is proposed in this paper, which not only reduces the computational overhead of the signer and the verifier, but also ensures the security of the server-aided verification phase by resisting the collusion attack of the signer and the server. The most important is that the scheme could verify the validity of partial signature generated by the server, so as to ensure the security of generation phase of the server-aided signature. Finally, our formal security analysis verifies the security of the SA-VABS scheme, and simulation experiments as well as comparative analysis indicate that the SA-VABS scheme improves security while ensuring efficiency.