Quantum Secure Clockwise Sorting

Secure computational geometry (SCG) is an important type of secure multi-party computation (SMC) problem, which studies how to calculate the relationship of several geometric objects securely. Quantum SCG (QSCG) can achieve higher security than classical SCG protocols, but the achievements of QSCG are still limited at present. In this paper, we define a new SCG problem: the secure clockwise sorting (SCS) problem, and propose a quantum protocol to solve it. We first propose a quantum secure clockwise comparison subprotocol, where the cross and scalar products of two vectors are calculated by using a quantum secure two-party scalar product protocol, and then the relative clockwise order of two points is determined. Based on the subprotocol, we then give a quantum SCS protocol to determine the clockwise order of a series of points. Finally, we show the correctness, security, and efficiency of our protocol through detailed performance analysis.