Efficient Quantum Secure Vector Dominance and Its Applications in Computational Geometry

Secure vector dominance is a key cryptographic primitive in secure computational geometry (SCG), determining the dominance relationship of vectors between two participants without revealing their private information. However, the security of traditional SVD protocols is compromised by the formidable computational power of quantum computing, and their efficiency needs further improvement. To address these challenges, an efficient quantum secure vector dominance (QSVD) protocol is proposed. Specifically, we first introduce a quantum private permutation (QPP) subprotocol to shuffle the elements of each participant's private input vector. To further facilitate secure data comparison, we propose an enhanced quantum millionaire subprotocol with equality determination functionality, building upon Jia's original protocol. Based on the above two subprotocols, we propose a QSVD protocol with polynomial complexity, deriving vector dominance in a single interaction with a semi-honest third party. Performance analyses confirm that QSVD protocol is correct, resilient against malicious attacks, and retains polynomial computational complexity, ensuring both security and efficiency. To demonstrate the scalability of the QSVD protocol, we illustrate its applications in several geometric computation problems, such as point-line inclusion determination, line-line intersect determination, and point-in-polygon determination. Finally, we validate the feasibility of our protocol by conducting comprehensive simulations on IBM's Qiskit platform, demonstrating its practical applicability and effectiveness in real quantum computing environments.