M. Dehghan - Department of Computer Engineering, Amirkabir University of Technology, Tehran, Iran
B. Sadeghiyan - Department of Computer Engineering, Amirkabir University of Technology, Tehran, Iran
E. Khosravian - Department of Mechanical Engineering, Payame Noor University, Tehran, Iran

In this paper, two protocols are presented for private intersection detection of two moving objects’ trajectories. Assuming that the movement trajectory of an object can be described by a time-dependent polynomial function, the problem of finding the intersection points is simplified to the problem of finding the common roots of the corresponding polynomials. Thereafter, GrÖbner Basis is used to design a novel secure protocol of finding the common roots of the polynomials. Another protocol is also designed based on the distance computation of two trajectories’ curves. Moreover, we present the complexity analysis of the protocol for private trajectory intersection testing of two moving objects, which is based on GrÖbner Basis. Then, we compare its complexity by the garbled circuit-based protocol for Euclidean Distance Computation of l points. We also prove the security of our proposed protocol, which is based on the distance computation of two curves. Keywords: Private Trajectory Intersection Testing, GrÖbner Basis, Distance Computation, Complexity , Garbled Circuit, Euclidean Distance

Private Trajectory Intersection Testing, GrÖbner Basis, Distance Computation, complexity, Garbled Circuit, Euclidean Distance