Satellites have long been reliable and widely used mediums for communication. With recent advancements in the Internet of Things (IoT), satellite-based IoT has emerged, enhancing global connectivity by enabling real-time communication between remote sensing devices and central hubs through satellite links. However, secure and authenticated communication between Low Earth Orbit (LEO) satellites and resource-constrained ground devices remains underexplored. This paper introduces a novel homomorphic signcryption scheme designed for satellite IoT environments. Our scheme employs hyperelliptic curve cryptography and homomorphic encryption to facilitate the privacy-preserving transmission of aggregated data from multiple IoT devices to LEO satellites. We benchmark our homomorphic signcryption scheme against several established schemes tailored for different network environments, including for satellite networks, Identity-based Signcryption for Smart Grids, 3-Factor Authentication schemes and heterogeneous Vehicular Ad-hoc Networks (VANETs) encryption schemes. Our scheme demonstrates substantial improvements in computational efficiency, communication overhead, and storage capacity. It reduces computational costs by 90% to 99%, communication overhead by 41% to 92%, and storage demands by 62% to 93% compared to existing methods. These quantitative findings demonstrate the practical viability of our approach for satellite IoT deployments with limited resources. Additionally, the robustness of our approach is confirmed through formal security validation using BAN Logic and Scyther, which verifies its strong security, optimal performance, and privacy preservation capabilities, making it ideally suited for real-world satellite IoT systems.