Exploring Differential Obliviousness

In a recent paper, Chan et al. [SODA \u2719] proposed a relaxation of the notion of (full) memory obliviousness, which was introduced by Goldreich and Ostrovsky [J. ACM \u2796] and extensively researched by cryptographers. The new notion, differential obliviousness, requires that any two neighboring inputs exhibit similar memory access patterns, where the similarity requirement is that of differential privacy. Chan et al. demonstrated that differential obliviousness allows achieving improved efficiency for several algorithmic tasks, including sorting, merging of sorted lists, and range query data structures. In this work, we continue the exploration of differential obliviousness, focusing on algorithms that do not necessarily examine all their input. This choice is motivated by the fact that the existence of logarithmic overhead ORAM protocols implies that differential obliviousness can yield at most a logarithmic improvement in efficiency for computations that need to examine all their input. In particular, we explore property testing, where we show that differential obliviousness yields an almost linear improvement in overhead in the dense graph model, and at most quadratic improvement in the bounded degree model. We also explore tasks where a non-oblivious algorithm would need to explore different portions of the input, where the latter would depend on the input itself, and where we show that such a behavior can be maintained under differential obliviousness, but not under full obliviousness. Our examples suggest that there would be benefits in further exploring which class of computational tasks are amenable to differential obliviousness

Zig-zag Sort: A Simple Deterministic Data-Oblivious Sorting Algorithm Running in O(n log n) Time

We describe and analyze Zig-zag Sort--a deterministic data-oblivious sorting algorithm running in O(n log n) time that is arguably simpler than previously known algorithms with similar properties, which are based on the AKS sorting network. Because it is data-oblivious and deterministic, Zig-zag Sort can be implemented as a simple O(n log n)-size sorting network, thereby providing a solution to an open problem posed by Incerpi and Sedgewick in 1985. In addition, Zig-zag Sort is a variant of Shellsort, and is, in fact, the first deterministic Shellsort variant running in O(n log n) time. The existence of such an algorithm was posed as an open problem by Plaxton et al. in 1992 and also by Sedgewick in 1996. More relevant for today, however, is the fact that the existence of a simple data-oblivious deterministic sorting algorithm running in O(n log n) time simplifies the inner-loop computation in several proposed oblivious-RAM simulation methods (which utilize AKS sorting networks), and this, in turn, implies simplified mechanisms for privacy-preserving data outsourcing in several cloud computing applications. We provide both constructive and non-constructive implementations of Zig-zag Sort, based on the existence of a circuit known as an epsilon-halver, such that the constant factors in our constructive implementations are orders of magnitude smaller than those for constructive variants of the AKS sorting network, which are also based on the use of epsilon-halvers.Comment: Appearing in ACM Symp. on Theory of Computing (STOC) 201

Privacy-preserving proximity detection with secure multi-party computational geometry

Over the last years, Location-Based Services (LBSs) have become popular due to the global use of smartphones and improvement in Global Positioning System (GPS) and other positioning methods. Location-based services employ users' location to offer relevant information to users or provide them with useful recommendations. Meanwhile, with the development of social applications, location-based social networking services (LBSNS) have attracted millions of users because the geographic position of users can be used to enhance the services provided by those social applications. Proximity detection, as one type of location-based function, makes LBSNS more flexible and notifies mobile users when they are in proximity. Despite all the desirable features that such applications provide, disclosing the exact location of individuals to a centralized server and/or their social friends might put users at risk of falling their information in wrong hands, since locations may disclose sensitive information about people including political and religious affiliations, lifestyle, health status, etc. Consequently, users might be unwilling to participate in such applications. To this end, private proximity detection schemes enable two parties to check whether they are in close proximity while keeping their exact locations secret. In particular, running a private proximity detection protocol between two parties only results in a boolean value to the querier. Besides, it guarantees that no other information can be leaked to the participants regarding the other party's location. However, most proposed private proximity detection protocols enable users to choose only a simple geometric range on the map, such as a circle or a rectangle, in order to test for proximity. In this thesis, we take inspiration from the field of Computational Geometry and develop two privacy-preserving proximity detection protocols that allow a mobile user to specify an arbitrary complex polygon on the map and check whether his/her friends are located therein. We also analyzed the efficiency of our solutions in terms of computational and communication costs. Our evaluation shows that compared to the similar earlier work, the proposed solution increases the computational efficiency by up to 50%, and reduces the communication overhead by up to 90%. Therefore, we have achieved a significant reduction of computational and communication complexity

Exploring Differential Obliviousness

In a recent paper Chan et al. [SODA '19] proposed a relaxation of the notion of (full) memory obliviousness, which was introduced by Goldreich and Ostrovsky [J. ACM '96] and extensively researched by cryptographers. The new notion, differential obliviousness, requires that any two neighboring inputs exhibit similar memory access patterns, where the similarity requirement is that of differential privacy. Chan et al. demonstrated that differential obliviousness allows achieving improved efficiency for several algorithmic tasks, including sorting, merging of sorted lists, and range query data structures. In this work, we continue the exploration and mapping of differential obliviousness, focusing on algorithms that do not necessarily examine all their input. This choice is motivated by the fact that the existence of logarithmic overhead ORAM protocols implies that differential obliviousness can yield at most a logarithmic improvement in efficiency for computations that need to examine all their input. In particular, we explore property testing, where we show that differential obliviousness yields an almost linear improvement in overhead in the dense graph model, and at most quadratic improvement in the bounded degree model. We also explore tasks where a non-oblivious algorithm would need to explore different portions of the input, where the latter would depend on the input itself, and where we show that such a behavior can be maintained under differential obliviousness, but not under full obliviousness. Our examples suggest that there would be benefits in further exploring which class of computational tasks are amenable to differential obliviousness

Oblivious Median Slope Selection

We study the median slope selection problem in the oblivious RAM model. In this model memory accesses have to be independent of the data processed, i.e., an adversary cannot use observed access patterns to derive additional information about the input. We show how to modify the randomized algorithm of Matou\v{s}ek (1991) to obtain an oblivious version with O(n log^2 n) expected time for n points in R^2. This complexity matches a theoretical upper bound that can be obtained through general oblivious transformation. In addition, results from a proof-of-concept implementation show that our algorithm is also practically efficient.Comment: 14 pages, to appear in Proceedings of CCCG 202