Abstract: To improve the computational efficiency of the distance between two three-dimension parallel straight lines, this paper proposes two protocols of a secure square of two real numbers sum protocol (SSTS) and Paillier homomorphic encryption algorithm (PHEA). Firstly, SSTS is to derive the distance between two three-dimension parallel straight lines by the distance from a point to a straight line, and it constructs auxiliary data to hide their own specific data by a secure square of two real numbers sum protocol. Secondly, PHEA uses Paillier homomorphic encryption algorithm to hide linear equation coefficient. Parties can communicate with each other, but will not reveal their specific data. Two protocols are confidential to calculate the distance between two three-dimension parallel straight lines. In addition, this paper proves the validity of two protocols, and proves the security by the simulation example. Finally, this paper analyzes the computation complexity and communication complexity of SSTS and PHEA with the existing protocol. The results show that the new protocols are at least 50% less than that of the existing protocol.

Key words: privacy-preserving computational geometry, space distance, secure square of two real numbers sum, homomorphic encryption, simulation paradigm

摘要： 为提高三维空间两平行直线间距离协议的计算效率，基于安全两实数和平方（secure square of two real numbers sum，SSTS）计算协议与Paillier同态加密算法（Paillier homomorphic encryption algorithm，PHEA）分别提出了三维空间两平行直线间的距离计算协议。SSTS协议利用空间任一点到直线的距离推导出三维空间两平行直线间的距离，通过安全两实数和平方计算协议构造辅助数据来隐藏自己的具体数据；PHEA协议通过Paillier同态加密算法将自己直线方程的系数隐藏，能与对方进行交流计算，但不会泄露自己的具体数据；两个协议均能保密地计算出三维空间两平行直线间的距离。分别证明了两个协议的正确性，并利用模拟范例证明了两个协议的安全性。最后，对SSTS协议和PHEA协议与现有协议进行比较分析，结果表明，新协议有较低的计算复杂性和通信复杂性，比现有协议至少降低了50%。

关键词: 隐私保护的计算几何, 空间距离, 安全两实数和平方, 同态加密, 模拟范例