## 2 Related Work  
In this section, we review existing related works in the domain of secure querying on spatio-temporal databases. Our discussion includes privacy-preserving approaches for trajectory-based queries. In recent years, trajectory databases have constituted an important research area that has received a lot of interest. Most researchers have focused on the querying of moving objects and their trajectory. The so-called trajectory-based queries are also gaining much interest. The queries based on trajectory data require the knowledge of the whole, or at least a part of the mobile objects' trajectory to be processed. Such queries may provide useful information about an object's average speed, travelled distance, and so forth. In [8], three common mechanisms in privacy-preserving trajectory publishing are described. Generalization and suppression are the most common ones used to implement the \(k\)-anonymity. However, the main drawback of these mechanisms is that they suffer from a high possibility of information loss--thus, perturbation techniques based on randomization (e.g., adding noise) may be utilized as an alternative. Actually, the problem of secure querying on spatio-temporal data in combination with \(k\)-anonymity has gained much attraction among researchers. Indeed, authors in [10] describe the historical \(k\)-anonymity based on each mobile user's trajectory data history, known as Personal History Locations (PHL). According to PHL anonymity, a user, U is camouflaged by \(k-1\) users whose PHLs have a common part with its own, rendering him/her indistinguishable among them. Privacy preservation is enforced as the generalization method has been applied. More specifically, by trying to preserve historical \(k\)-anonymity, authors increased the uncertainty related to the user's real location data at the time of the query by modifying the spatio-temporal information of the query. More precisely, in [11], by employing the \(k^{l}\) anonymity privacy model, authors ensure that an intruder, who has knowledge of any sub-trajectory \(TS\) of size \(l\) of a user's trajectory \(T^{j}\), cannot distinguish their one among \(k-1\) trajectories that protect them with probability, based on \(TS\), at most \(\frac{1}{k}\). In a more recent work [9], the authors investigated the privacy-preserving problem based on real spatio-temporal data. That paper employed the \(k\)-anonymity method and formed the anonymity set based on motion vectors with the aim of executing secure spatial \(k\)-NN queries. More specifically, the problem of \(k\)-anonymity from a dimensionality perspective and the impact of used dimensions on the vulnerability of suggested methods was investigated. The experiments presented the effectiveness of the proposed method, such as the clustering under particular attributes combination, and observed that it benefited from attributes suppression during the \(k\)-anonymity set computation. Authors in [12] suggested a novel spatio-temporal Mysql ReTrieval framework based on the MySQL and PostgreSQL database management system. In the context of that work, authors employed Hough-X transformation so as to evaluate the efficiency of range queries on nonlinear two-dimensional trajectories of mobile objects. Indeed, they demonstrated that the Hough-X dual approach, in combination with the range-tree variant, was quite efficient. Generally, the trajectory of a mobile user is non-linear. However, it can be approximated by a discrete number of linear sub-trajectories with the use of a trajectory segmentation application. Each partition is represented by a line segment between two consecutive partition points, and is expected to provide an effective and efficient way to obtain insights into motion characteristics and behavioral preferences of mobile objects. Our approach performs low-rate sampling and considers linear interpolation between successive sampled points, where each line segment represents the continuous moving of the object during sampled points.