With the widespread application of the Internet of Things (IoT), ensuring communication security for IoT devices is of considerable importance. Since IoT data are vulnerable to eavesdropping, tampering, forgery, and other attacks during an open network transmission, the integrity and authenticity of data are fundamental security requirements in the IoT. A certificateless signature (CLS) is a viable solution for providing data integrity, data authenticity, and identity identification in resource-constrained IoT devices. Therefore, designing a secure and efficient CLS scheme for IoT environments has become one of the main objectives of IoT security research. However, the existing CLS schemes rarely focus on strong unforgeability and replay attacks. Herein, we design a novel CLS scheme to protect the integrity and authenticity of IoT data. In addition to satisfying the strong unforgeability requirement, the proposed scheme also resists public key replacement attacks, malicious-but-passive key-generation-centre attacks, and replay attacks. Compared with other related CLS schemes without random oracles, our CLS scheme has a shorter private key, stronger security, and lower communication and computational costs.