IJE TRANSACTIONS A:Basics Vol. 34, No. 04, (April 2021) 863-872 Please cite this article as: M. Dehghan, B. Sadeghiyan, E. Khosravian, Private Trajectory Intersection Testing: Is Garbled Circuit Better than Custom Protocols? International Journal of Engineering, Transactions A:Basics Vol. 34, No. 04, (2021) 863-872 International Journal of Engineering Journal Homepage: www.ije.ir Private Trajectory Intersection Testing: Is Garbled Circuit Better than Custom Protocols? M. Dehghan* a , B. Sadeghiyan a , E. Khosravian b a Department of Computer Engineering, Amirkabir University of Technology, Tehran, Iran b Department of Mechanical Engineering, Payame Noor University, Tehran, Iran PAPER INFO Paper history: Received 14 July 2020 Received in revised form 07 January 2021 Accepted 14 January 2021 Keywords: Distance Computation Euclidean Distance Garbled Circuit GrÖbner Basis Private Trajectory Intersection ABSTRACT In this paper, two protocols are presented for private intersection detection of two moving objects’ trajectories. To design the first protocol, we simplify the problem of finding the intersection points to the problem of finding the common roots of the polynomials, which represent the moving objects’ trajectories. Thereafter, GrÖbner Basis is used to design a novel secure protocol to find the common roots of the polynomials. Another protocol is also designed based on the distance computation of two trajectories’ curves. The complexity of the GrÖbner-based protocol for finding the common roots of polynomials is numerical. Its complexity is much lower than the complexity of the garbled circuit- based protocol for Euclidean Distance Computation of l points and the complexity of the protocols for private proximity testing. 1. INTRODUCTION 1 Private Trajectory Intersection Testing (PTIT) can be considered as a problem in which some moving objects wish to detect the intersection of their trajectories. However, they intend to keep their trajectories secret, namely, to detect whether the distance between moving objects’ trajectories is larger than d (d is a predefined threshold), while the parties do not reveal their trajectories of movement. The problem of PTIT can have some applications in many scenarios, such as urban traffic management, RoboCup competitions, aircraft ad hoc networks, mobile networks and etc. Suppose that the drivers wish to manage traffic in a way to prevent congestion and distribute traffic in streets and highways without revealing their trajectories. Therefore, they want to use a PTIT Protocol. In addition, in RoboCup competitions, the participants tend to select the best trajectories for their robots and do not collide with other robots. These participants apply a PTIT protocol without revealing their trajectories. *Corresponding Author Institutional Email: motahareh479@aut.ac.ir (M. Dehghan) The “privacy” is generally in conflict with collecting, storing, using, processing and sharing of personally identifiable data. The primary objective of privacy measures is to ensure proper protection of private data in the course of processing or dissemination of sensitive information [1]. The main feature of PTIT protocol is privacy preserving of moving objects’ trajectories. Secure multiparty computation is an approach to support privacy. In this approach, a set of parties with secret inputs wish to compute some joint functions of their inputs, while they wish to preserve the privacy property. The first general solution for the problem of secure two-party computation in the presence of semi- honest adversaries was presented by Yao [2]. Later, solutions were provided for the multi-party and malicious adversarial cases by Goldreich et al. [3]. In 2016, Hemenway et al. computed the probability of intersection between satellites’ trajectories [4]. In their work, the approach is based on garbled circuits, where the garbled circuit for detection of intersection is so large and has not good performance. Atallah and Du doi: 10.5829/ije.2021.34.04a.12