With the increasing adoption of connected and autonomous vehicles, platooning services have emerged as a promising solution to enhance road traffic efficiency. However, the widespread deployment of platooning services raises concerns about the privacy and security of sensitive vehicle data. This paper proposes a novel privacy-preserving framework for platoon formation, named H3PC (Homomorphic Privacy-Preserving Platooning Construction), to address privacy and security challenges. The proposed approach is based on fully homomorphic encryption and order-preserving encryption, to enable secure and private operations within the platoon construction. In addition, the H3PC framework incorporates a safe vehicle control method that adheres to established norms in the literature. By striking a balance between security and computational efficiency, H3PC enables effective platooning construction. Furthermore, we present experimental results demonstrating the performance and latency of H3PC.