Smart villages, utilizing digital technology, present a viable solution to address rural communities' needs, especially in developing countries. However, high infrastructure costs, informal economies, and governance issues pose significant challenges, impeding the realization of government aspirations for the smart village concept. This study focuses on preserving privacy and enhancing security within IoT-enabled smart villages. We propose a lightweight, privacy-preserving remote user authentication protocol utilizing geometric secret sharing to mitigate security vulnerabilities in resource-constrained IoT devices. The protocol eliminates the need for traditional passwords and smart cards, reducing user burden and enhancing usability. Performance analysis and comparative evaluations, including power consumption, security features, and computational costs, demonstrate the efficacy and robustness of our protocol.