The satellite Internet of Things (satellite IoT) has the characteristics of large space-time span and highly open communication links. While effectively expanding the spatial capability of the traditional Internet of Things, it will face security threats such as impersonation, replay, tampering and eavesdropping of the traditional Internet of Things and satellite communication. In this paper, an SM2-based certificateless integrated signature and encryption scheme (SM2-CL-ISE) is proposed for satellite IoT with key optimization and conditional anonymity. Then incorporating Geostationary Earth Orbit (GEO) satellite, a Low Earth Orbit (LEO) satellite authentication protocol and a static terminal device authentication protocol are designed. In addition, we prove the security of SM2-CL-ISE under the formal security model, and further discuss how the proposed authentication schemes can satisfy those essential security requirements. To evaluate the effectiveness of our proposed protocols, we conducted several experiments and compared their performance with that of existing protocols. The experimental results show that our scheme achieves more efficient performance with a slightly increased communication overhead on authentication.