PTreeB: Efficient and Privacy-Preserving k-d Tree Building over Vertically Distributed Data in Cloud

Big data have witnessed a growing trend towards vertically distributed storage, with various queries on vertically organized data recognized as effective means for unlocking data's inherent value. Several solutions have emerged for enabling privacy-preserving queries on vertically distributed data using secure multi-party computation techniques. However, these approaches often involve substantial communication overheads among data owners and place significant computational burdens on them, rendering them impractical for resource-constrained data owners. Outsourcing vertically distributed queries to the cloud can substantially alleviate the computational burdens on data owners, and efficient index construction is crucial for outsourced queries on vertical data. In light of this, we present the pioneering “Privacy-Preserving k-d Tree Building” (PTreeB) scheme for vertically distributed outsourced data in this study. Our scheme begins with the development of a private random dimension choosing algorithm (PCDim) and a private equality test (PET) algorithm, leveraging additive Paillier homomorphic encryption. Subsequently, these algorithms, along with various efficiency-enhancing strategies, including pre-sorting each data owner's data and adopting a dual-key system for data privacy protection, form the foundation of our PTreeB scheme. We rigorously demonstrate the security of our scheme, and its efficiency is validated through extensive experimentation.