## 3 Materials and Methods
### Privacy Preservation Analysis  
Privacy relates to individual data protection and the human right to be able to determine the information about themselves that is to be hidden. Privacy-preserving data management includes \(k\)-anonymity, a noted method for data anonymization before publication, which has also been studied in the context of trajectory data. Authors in [16] claim that given a set of trajectories, the objective of the data publication is to transform them into some \(k\)-anonymized form in order to prevent original data publication, putting at risk the privacy of individuals related with the data. In addition, they mention that an intruder, who knows a sub-trajectory of the original trajectory of an individual, may utilize it with the aim of extracting the whole trajectory of that person based on the published data. Finally, they recognize an upper bound for the re-identification probability of the whole trajectory within the released data, namely \(1/k\), where the parameter \(k\) reflects the expected level of privacy. Our solution transforms the original spatial point into the dual-point one using bijective mapping, such as Hough. This technique allows for a \(k\)-NN search directly on the transformed points, thus providing stronger location privacy. Assuming an insecure Transformed Database Management System (TDBMS) possibly located at a third party (e.g., a service provider in the cloud), an attacker sees its environment. In particular, the attacker has access to the transformed database, to the queries upon the transformed data, as well as to the results. Also, we suppose that the attacker is aware of the dual transformation scheme and aims to retrieve the original database executing Hough-X and/or Hough-Y algorithms with respect to the size of the database. Nonetheless, in our proposed paper, we aim to prevent an attacker from obtaining the original database, as it is possible that they may occupy  
Figure 3: Theoretical curve of compression ratio for M = [10 100 1000 10,000 100,000]. extra knowledge about this original database. To better evaluate the power of the transformation scheme, we taxonomize the attacks into different levels based on the possessed knowledge. 1. **Level 1**: The attacker only observes the transformed database. 2. **Level 2**: Except the transformed database, the attacker is familiar with a set of plain tuples of the original database, but does not know the corresponding encoded values of those tuples in the transformed database. 3. **Level 3**: Apart from the transformed database, the attacker observes a set of tuples in the original database, and thus knows the corresponding encoded values of those tuples. A few cryptography-based approaches, such as homomorphic encryption (HE), verifiable computation (VC), and secure multi-party computation (MPC) have been designed in order to provide secure big-data processing in the Cloud [17]. However, other approaches, such as Asymmetric Key Cryptography and trusted Public Key Infrastructure have been developed over the years in order to support privacy preservation in the spatio-temporal domain. The basic idea behind these techniques is to encrypt the identity of the user prior to sending it to the service provider. In this way, the service provider does not have any knowledge about the real identity of the individual who initiated the \(k\)-NN query. To prevent an external adversary from linking queries to the same mobile object, its pseudonym has to be secure. For this reason, we are concerned about pseudonyms' recovery, as well as registration protocols consisting of three entities, namely, Users (U), Identity Provider (IP), and Service Providers (SP). Recall that they are based on Brand's credentials and have been suggested by Brand in the context of \"The New System\" with the aim of making the communication more reliable and secure. We believe that the adoption of these protocols will reinforce the identity privacy of mobile objects and the spatio-temporal databases at large.