Privacy-preserving two-party computation of line segment intersection

By considering maps and routes as sequences of line segments, their intersections can be computed to find out useful information like the possibility of collision in a military area where the parties do not trust each other. At the first glance, finding the coordinates of the intersections is seemed impossible to be solved securely since having coordinates of two intersection points on the same line reveals the passing line. In this paper, we solve this problem by suggesting a secure two-party protocol in presence of passive adversaries. Additionally, regarding the fact that in some cases, the fixedness of the inputs of the parties in classic security models is an unrealistic assumption, we define the new concept of input-adaptive security and show that our method is secure against such an adversary who is able to select his inputs adaptively. In addition to serve different approaches like oblivious transfer and sometimes homomorphic encryption, we also employ some tricks to prevent the distribution of harmful information between specific parties to achieve our intended security level. We provide formal proofs to show the security of our protocol. Time complexity analysis and implementations show that our protocol finds the intersections in feasible time of O(nlogn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}}(n \log n)$$\end{document} and indicate that our protocol is as good as the unsecure optimal method of line segment intersection computation. In comparison, previous methods require O(n2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(n<^>2)$$\end{document} to only detect the existence of intersection between two sets of n line segments and are unable to find the coordinates of the intersections.