Privacy-preserving computational geometry is the research area on the intersection of the domains of secure multi-party computation (SMC) and computational geometry. As an important field, the privacy-preserving geometric intersection (PGI) problem is when each of the multiple parties has a private geometric graph and seeks to determine whether their graphs intersect or not without revealing their private information. In this study, through representing Alice's (Bob's) private geometric graph G_A (G_B) as the set of numbered grids S_A (S_B), an efficient privacy-preserving quantum two-party geometric intersection (PQGI) protocol is proposed. In the protocol, the oracle operation O_A (O_B) is firstly utilized to encode the private elements of S_A=(a_0, a_1, ..., a_(M-1)) (S_B=(b_0, b_1, ..., b_(N-1))) into the quantum states, and then the oracle operation O_f is applied to obtain a new quantum state which includes the XOR results between each element of S_A and S_B. Finally, the quantum counting is introduced to get the amount (t) of the states |a_i+b_j> equaling to |0>, and the intersection result can be obtained by judging t>0 or not. Compared with classical PGI protocols, our proposed protocol not only has higher security, but also holds lower communication complexity.

翻译：隐私保护的计算几何是安全多方计算与计算几何交叉领域的重要研究方向。作为该领域的关键问题，隐私保护几何交集问题描述为：多方各自持有私有几何图形，需在不泄露各自隐私信息的前提下判定这些图形是否存在交集。本研究通过将Alice（Bob）的私有几何图形G_A（G_B）表示为编号网格集合S_A（S_B），提出了一种高效的隐私保护量子两方几何交集协议。该协议首先利用Oracle操作O_A（O_B）将S_A=(a_0, a_1, ..., a_(M-1))（S_B=(b_0, b_1, ..., b_(N-1))）的私有元素编码至量子态，随后通过Oracle操作O_f获得包含S_A与S_B各元素间异或结果的新量子态，最后引入量子计数算法统计|a_i+b_j>=|0>的态数量t，并通过判断t>0与否得出几何交集结论。与经典PGI协议相比，本协议不仅具备更高的安全性，还拥有更低的通信复杂度。