Cooperative Multihop Localization with Privacy Golaleh Rahmatollahi Institute of Comm. Technology Leibniz University of Hannover Appelstr. 9a, 30167 Hannover, Germany gola@ikt.uni-hannover.de Stefano Severi and Giuseppe Abreu School of Science and Engineering, Jacobs University Bremen, Germany Campus Ring 1, 28759 Bremen g.abreu@jacobs-univeristy.de Abstract—We introduce a cooperative algorithm for self and target network localization with privacy. The algorithm differs form other cooperative localization algorithms in which it does not require nodes to disclose their location or even to measure (or share) their mutual distances. This is achieved by a combination of two factors: a) a novel closed-form statistical relationship between the hop- and Euclidean-distances of distributed random Breadth Search First (BSF) paths; and b) novel multihop local- ization algorithms. The results, compared against conventional multihop distance collection indicate that, remarkably, the pri- vacy offered by the proposed cooperative localization algorithm does not incur any significant sacrifice in accuracy. I. I NTRODUCTION In distributed cooperative localization, sensor nodes being unaware of their location cooperate with each other by ex- changing mutual distances or positions and relaying packets to nodes with a priori location information referred to as anchors. The algorithms currently considered for cooperative local- ization can be basically classified into so-called message- passing localization algorithms (MPLA)’s and range-based multihop localization algorithms (MHLA)’s. The first ap- proach relies on iterative methods whereby each node com- putes either its own location and an associated confidence, or an entire probability density function of its location, and passes that information to its neighbors [1], [2]. Since MPLA’s basically require all nodes to perform localization simultane- ously [3], some characteristic problems of these algorithms are: i) a substantial communication complexity, in terms of the amount of message-exchange required for convergence, which grows geometrically with the node-count N in the network [4]; ii) uncertainty in terms of convergence, since later requires a certain ratio between anchors and target nodes; iii) sensitivity to network topology changes due to mobility; and iv) an inherent lack of privacy, since nodes need to fully disclose their location to neighbors. In comparison, in MHLA’s the process of estimating po- sitions involves only one source node s, and a fixed number of anchor nodes (3 in 2D, 4 in 3D, at least) which can be reached by s via multiple hops [5]–[8]. Since MHLA are very robust to source-to-anchor distance estimation errors, accurate multihop localization is possible even if the number of hops is large [7], [8]. Furthermore, anchors can be found efficiently and fast through network discovery procedures such as the distributed Breadth First Search (BFS) algorithm [4] which require far less message exchanges and are more robust to network changes. But besides the issues of communication complexity, robust- ness to mobility and convergency, privacy is another potential advantage of MHLA’s over MPLA’s, which follows from the fact that in MHLA’s nodes need not to disclose their location with neighbors. In order to ensure full privacy, however, it is required that nodes do not share even their mutual distances. In this article we break through that last frontier by in- troducing a hop-count based MHLA with full privacy. This is achieved by presenting a novel closed-form statistical relation- ship between the number of hops and the separation distance of two nodes connected via BFS-like paths. In particular, distance estimation between source node and an anchor is obtained by cooperation in terms of relaying packets and counting the number of hops N H . To be evaluated, the expressions require only a pair of parameters, namely the network node density λ and the average hop length ¯ d which accounts for routing strategy (in the case in question, BFS branches). Using these expressions, multihop distances can be accu- rately estimated with basis on simple hop-counts, and ap- plied to novel and robust MHLA’s, namely, the Constrained Semidefinite Programming (CSDP) and Distance Contraction (DC) algorithms [7], [8], in order to estimate the target’s position. We will show that the proposed distributed cooperative localization scheme, while maintaining full privacy, does not incur any significant sacrifice in accuracy compared to existing alternatives where multihop distances are collected also by performing ranging between all pairs of nodes in the route. The article is organized as follows. In section II, we briefly describe the multihop network localization scenario and two multihop localization algorithms. The novel closed- form statistics on hop-count distribution is derived in section III. In section IV, the design of a distributed cooperative localization algorithm with privacy on the basis of the novel hop-count distribution is provided, and its performance is compared to “conventional” multihop localization schemes. Finally, conclusions are offered in section V. II. MULTIHOP LOCALIZATION ALGORITHMS A. Network and Localization Scenario A typical realization of a 2D multihop network consists of N nodes with density λ uniformly deployed with coordinates denoted by θ  [θ 0 ... θ N ]. We assume that only a small fraction of nodes is aware of their location. These special nodes are referred to as anchors and their coordinates are denoted by A =[a 1 ... a Na ], with N a > 2. Without loss of generality, we focus on the localization of a source s located at θ 0 close to the center of a circle formed by randomly placed anchors with separation distance D 0i . Since for localization purpose it is required that a source node is connected to the anchors via multihopping the node density λ has to be sufficiently large. Bounds on this required critical density λ ∗ have been found using tools of percolation theory [9]–[11]. The sharpest pair of analytical lower and upper bounds currently known are due to Kong et al. [11] λ ∗ L ≈ 0.7698 and Hall [9] λ ∗ U ≈ 0.8426.