Quantum‐Secure Point‐Circle, Circle‐Circle Relation Determination Protocol With Application to Privacy‐Preserving Geometric Range Query

As an important branch of quantum‐secure multi‐party computation, quantum‐secure multi‐party computational geometry aims to solve geometric computation problems without revealing the parties' geometric inputs. Privacy‐preserving point‐circle and circle‐circle positional relationships are classic problems in secure multi‐party computational geometry, with broad applications in privacy‐preserving queries, collision detection, and clustering. Building on the quantum two‐party summation and multiplication protocols proposed by Sutradhar and Om, this paper presents the first quantum‐secure protocols for determining point‐circle and circle‐circle positional relationships, enabling accurate relationship determination without disclosing private geometric parameters. Security analysis and simulation experiments on the IBM Qiskit platform demonstrate the security and feasibility of the proposed protocols. Furthermore, since existing quantum privacy‐preserving range query protocols are limited to one‐dimensional linear spaces, we build on the proposed protocols to present the first quantum‐secure privacy‐preserving geometric range query protocol, which enables arbitrary geometric range search for private coordinates in the two‐dimensional space while protecting the privacy of the query range and data points.