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In this paper, we present secure protocols for obtaining the collision points of two or more moving objects, while preserving their movements' privacy. We assume that the paths of moving objects can be presented with polynomial functions of time. We reduce the problem of obtaining collision points of two moving objects to the problem of obtaining common roots of polynomials. Then, we propose a secure protocol for obtaining the common roots of two polynomials, which is based on a Gröbner basis. Moreover, we propose another secure protocol for obtaining collision points of more than two moving objects based on a line segment intersection algorithm. We also demonstrate the security proof of our proposed protocols based on the ideal/real simulation paradigm.

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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 circuitbased protocol for Euclidean Distance Computation of l points and the complexity of the protocols for private proximity testing.