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## Section: New Results

### Foundations of information hiding

Information hiding refers to the problem of protecting private information while performing certain tasks or interactions, and trying to avoid that an adversary can infer such information. This is one of the main areas of research in Comète; we are exploring several topics, described below.

#### Differential privacy with general metrics.

Differential privacy can be interpreted as a bound on the distinguishability of two generic databases, which is determined by their Hamming distance: the distance in the graph determined by the adjacency relation (two databases are adjacent if they differ for one individual).

In [21] we lifted the restriction relative to the Hamming graphs and we explored the implications of differential privacy when the indistinguishability requirement depends on an arbitrary notion of distance. We showed that we can express, in this way, (protection against) kinds of privacy threats that cannot be naturally represented with the standard notion. We gave an intuitive characterization of these threats in terms of Bayesian adversaries, which generalizes the characterization of (standard) differential privacy from the literature. Next, we revisited the well-known result on the non-existence of universally optimal mechanisms for any query other than counting queries. We showed that in our setting, for certain kinds of distances, there are many more queries for which universally optimal mechanisms exist: Notably sum, average, and percentile queries. Finally, we showed some applications in various domains: statistical databases where the units of protection are groups (rather than individuals), geolocation, and smart metering.

#### Privacy for location-based services.

The growing popularity of location-based services, allowing unknown/untrusted servers to easily collect and process huge amounts of users' information regarding their location, has recently started raising serious concerns about the privacy of this kind of sensitive information. In [19] we studied geo-indistinguishability, a formal notion of privacy for location-based services that protects the exact location of a user, while still allowing approximate information - typically needed to obtain a certain desired service - to be released.

Our privacy definition formalizes the intuitive notion of protecting the user's location within a radius r with a level of privacy that depends on r. We presented three equivalent characterizations of this notion, one of which corresponds to a generalized version [21] of the well-known concept of differential privacy. Furthermore, we presented a perturbation technique for achieving geo-indistinguishability by adding controlled random noise to the user's location, drawn from a planar Laplace distribution. We demonstrated the applicability of our technique through two case studies: First, we showed how to enhance applications for location-based services with privacy guarantees by implementing our technique on the client side of the application. Second, we showed how to apply our technique to sanitize location-based sensible information collected by the US Census Bureau.

#### Relation between differential privacy and quantitative information flow.

Differential privacy is a notion that has emerged in the community of statistical databases, as a response to the problem of protecting the privacy of the database's participants when performing statistical queries. The idea is that a randomized query satisfies differential privacy if the likelihood of obtaining a certain answer for a database $x$ is not too different from the likelihood of obtaining the same answer on adjacent databases, i.e. databases which differ from $x$ for only one individual.

In [13] , we analyzed critically the notion of differential privacy in light of the conceptual framework provided by the Rényi min information theory. We proved that there is a close relation between differential privacy and leakage, due to the graph symmetries induced by the adjacency relation. Furthermore, we considered the utility of the randomized answer, which measures its expected degree of accuracy. We focused on certain kinds of utility functions called “binary”, which have a close correspondence with the Rényi min mutual information. Again, it turns out that there can be a tight correspondence between differential privacy and utility, depending on the symmetries induced by the adjacency relation and by the query. Depending on these symmetries we can also build an optimal-utility randomization mechanism while preserving the required level of differential privacy. Our main contribution was a study of the kind of structures that can be induced by the adjacency relation and the query, and how to use them to derive bounds on the leakage and achieve the optimal utility.

#### A differentially private mechanism of optimal utility for a region of priors

Differential privacy (already introduced in the previous sections) is usually achieved by using mechanisms that add random noise to the query answer. Thus, privacy is obtained at the cost of reducing the accuracy, and therefore the utility, of the answer. Since the utility depends on the user's side information, commonly modeled as a prior distribution, a natural goal is to design mechanisms that are optimal for every prior. However, it has been shown in the literature that such mechanisms do not exist for any query other than counting queries.

Given the above negative result, in [22] we considered the problem of identifying a restricted class of priors for which an optimal mechanism does exist. Given an arbitrary query and a privacy parameter, we geometrically characterized a special region of priors as a convex polytope in the priors space. We then derived upper bounds for utility as well as for min-entropy leakage for the priors in this region. Finally we defined what we call the tight-constraints mechanism and we discussed the conditions for its existence. This mechanism has the property of reaching the bounds for all the priors of the region, and thus it is optimal on the whole region.

#### Compositional analysis of information hiding

Systems concerned with information hiding often use randomization to obfuscate the link between the observables and the information to be protected. The degree of protection provided by a system can be expressed in terms of the probability of error associated to the inference of the secret information. In [14] we considered a probabilistic process calculus to specify such systems, and we studied how the operators affect the probability of error. In particular, we characterized constructs that have the property of not decreasing the degree of protection, and that can therefore be considered safe in the modular construction of these systems. As a case study, we applied these techniques to the Dining Cryptographers, and we derived a generalization of Chaum's strong anonymity result.

In [26] , a similar framework was proposed for reasoning about the degree of differential privacy provided by such systems. In particular, we investigated the preservation of the degree of privacy under composition via the various operators. We illustrated our idea by proving an anonymity-preservation property for a variant of the Crowds protocol for which the standard analyses from the literature are inapplicable. Finally, we made some preliminary steps towards automatically computing the degree of privacy of a system in a compositional way.

#### Preserving differential privacy under finite-precision semantics

The approximation introduced by finite-precision representation of continuous data can induce arbitrarily large information leaks even when the computation using exact semantics is secure. Such leakage can thus undermine design efforts aimed at protecting sensitive information. For instance, the standard approach to achieve differential privacy (introduced in previous sections) is the addition of noise to the true (private) value. To date, this approach has been proved correct only in the ideal case in which computations are made using an idealized, infinite-precision semantics. In [23] , we analyzed the situation at the implementation level, where the semantics is necessarily finite-precision, i.e. the representation of real numbers and the operations on them are rounded according to some level of precision. We showed that in general there are violations of the differential privacy property, and we studied the conditions under which we can still guarantee a limited (but, arguably, totally acceptable) variant of the property, under only a minor degradation of the privacy level. Finally, we illustrated our results on two cases of noise-generating distributions: the standard Laplacian mechanism commonly used in differential privacy, and a bivariate version of the Laplacian recently introduced in the setting of privacy-aware geolocation.

#### Metrics for differential privacy in concurrent systems

Many protocols for protecting confidential information have involved randomized mechanisms and a nondeterministic behavior (such as the Dining Cryptographers protocol or the Crowds protocol). In [28] , we investigate techniques for proving differential privacy in the context of concurrent systems which contain both probabilistic and nondeterministic behaviors. Our motivation stems from the work of Tschantz et al., who proposed a verification method based on proving the existence of a stratified family of bijections between states, that can track the privacy leakage, ensuring that it does not exceed a given leakage budget. We improve this technique by investigating state properties which are more permissive and still imply differential privacy. We consider three pseudometrics on probabilistic automata: The first one is essentially a reformulation of the notion proposed by Tschantz et al. The second one is a more liberal variant, still based on the existence of a family of bijections, but relaxing the relation between them by integrating the notion of amortization, which results into a more parsimonious use of the privacy budget. The third one aims at relaxing the bijection requirement, and is inspired by the Kantorovich-based bisimulation metric proposed by Desharnais et al. We cannot adopt the latter notion directly because it does not imply differential privacy. Thus we propose a multiplicative variant of it, and prove that it is still an extension of weak bisimulation. We show that for all the pseudometrics the level of differential privacy is continuous on the distance between the starting states, which makes them suitable for verification. Moreover we formally compare these three pseudometrics, proving that the latter two metrics are indeed more permissive than the first one, but incomparable with each other, thus constituting two alternative techniques for the verification of differential privacy.