Nowadays, the Internet of Things (IoT) and cloud computing have become more pervasive in the context of the industry as digitization becomes a business priority for various organizations. Therefore, industries outsource their crowdsourced Industrial IoT (IIoT) data in the cloud in order to reduce the cost for sharing data and computation. However, the privacy of such crowdsourced data in this environment has attracted wide attention across the globe. Signcryption is the significant cryptographic primitive that meets both requirement of authenticity and confidentiality of crowdsourced data among users/industries, and thus, it is ideal for ensuring secure authentic data storage and transmission in industrial crowdsourcing environments. In this paper, we introduce a new identity-based signcryption (IBSC) scheme using bilinear pairing for IIoT deployment. Besides, two hard problems are studied, called as, modified bilinear Diffie-Hellman inversion (MBDHI) assumption and modified bilinear strong Diffie-Hellman (MBSDH) assumption. The rigorous security analysis demonstrates that our IBSC scheme for IIoT is provably secure based on the intractability of decisional-MBDHI and MBSDH assumptions under formal security model without considering the concept of the random oracle. The performance comparison with other signcryption schemes shows satisfactory results. Thus, our IBSC scheme is appropriate for IIoT crowdsourcing environments, and also applicable for low-bandwidth communications.