Toward Efficient and Secure Hypercube Tree Building for Vertically Distributed Data in Cloud

The rapid development of big data and Internet of Things has promoted the formation of data silos, and cloud computing has facilitated the outsourcing of vertically distributed data to cloud servers. In outsourced query scenarios, building query indexes is crucial for balancing data utility and data privacy protection. The hypercube tree is a widely used index for multi-dimensional data, supporting various query types. Although secure hypercube tree-based queries have been extensively studied in existing works, they are not applicable for building a hypercube tree over vertically distributed ciphertext data. To address this issue, we propose the first efficient and secure hypercube tree building scheme for vertically distributed data, named SCTBuild. We first design a flexible three-party secret sharing (fTPSS) scheme, allowing data owners to flexibly configure secret sharing forms based on real-world computational, communication, and storage constraints. Then, we design a communication-efficient data outsourcing algorithm, a secure data permutation algorithm, and a secure data comparison algorithm based on the fTPSS scheme. After that, we propose our SCTBuild scheme based on the aforementioned algorithms, in which data owners first perform pre-computation on their data to improve tree-building efficiency. We prove that our fTPSS scheme, private algorithms, and the SCTBuild scheme are semantically secure in the simulation-based real and ideal worlds security model; and conduct experiments to validate their high efficiency.