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for a malicious anchor to determine the location of the request coming in (by using techniques such as time-of- ight) and send out the normal beacon if the location corresponds to an anchor node This also assumes that a malicious anchor node is aware of the locations of other anchor nodes The authors in this work ignore this, though, by assuming that a malicious anchor node would not be able to differentiate between the requests coming from an anchor node or a nonanchor node Another approach for dealing with malicious anchor nodes is given in [170] They present two different ideas to deal with this problem The rst is based on the observation that every node typically uses the mean square error calculation to estimate its location The outliers can then be ltered using the calculated error To exploit this observation a node rst determines its location using the MMSE-based method and information from all the anchors it hears from The node then assesses if the estimated location is consistent with the information transmitted by all the anchors If the estimated distance is consistent then the process terminates, else the node attempts to identify and remove the most inconsistent reference and repeats the process The process continues until either a set of consistent location references is obtained or a conclusion is reached that such a set is impossible to nd (since the number of location references is less than three) Given a set L of anchors and a threshold to be used to determine inconsistency, a naive approach to computing the largest set of consistent location references is to check all subsets of L starting with the entire set This could be done until a subset of L is found that is consistent or it can be concluded that it is not possible to nd such a subset As is obvious, this will be computationally inef cient, which is a big disadvantage given the resource constraints on nodes in ad hoc networks To address this, the authors in [170] also propose a greedy algorithm which works in rounds In the rst round, the set of all location references is used to verify that they are consistent If these are consistent then the algorithm outputs the estimated location and terminates Otherwise, the algorithm considers all subsets of location references with one fewer location reference than in the previous round and chooses the subset with the least mean square error as the input to the next round The algorithm continues until a subset is found that is consistent or it can be concluded that it is not possible to nd such a subset The second idea [170] is to use an iteratively re ned voting scheme in order to tolerate the malicious location information The target eld is divided into a grid of cells Each node determines the likelihood of being present in each cell using the information in the beacons received by it Each cell in which the sensor node can be present is given a vote Finally, the cell(s) with the highest votes is selected and the center of this cell(s) is the estimated location Note that both these ideas assume that the number of benign beacons is more than that of the malicious beacons Thus, in order to defeat these schemes, an attacker will have to ensure that the number of malicious beacon signals is more than the number of benign beacon signals Another way for the adversary to defeat these schemes is by ensuring that he is not too aggressive Note that this might result in some falsi cation of the location determination, which might not be signi cant Another approach that does not consider any speci c attacks but rather focuses on ensuring that statistical robustness is introduced in the computation phase of the localization process is taken in [171] The advantage of this approach is that it achieves robustness against novel and unforeseen attacks The authors illustrate here that the impact of outliers can be limited by employing a least median squares (LMS) technique Note
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