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Secure distance-based localization in the presence of cheating beacon (or anchor) nodes is an important problem in mobile wireless ad hoc and sensor networks. Despite significant research efforts in this direction, some fundamental questions still remain unaddressed: In the presence of cheating beacon nodes, what are the necessary and sufficient conditions to guarantee a bounded error during a two-dimensional distance-based location estimation? Under these necessary and sufficient conditions, what class of localization algorithms can provide this error bound? In this paper, we attempt to answer these and other related questions by following a careful analytical approach. Specifically, we first show that when the number of cheating beacon nodes is greater than or equal to a given threshold, there do not exist any two-dimensional distance-based localization algorithms that can guarantee a bounded error. Furthermore, when the number of cheating beacons is below this threshold, we identify a class of distance-based localization algorithms that can always guarantee a bounded localization error. Finally, we outline three novel distance-based localization algorithms that belong to this class of bounded error localization algorithms. We verify their accuracy and efficiency by means of extensive simulation experiments using both simple and practical distance estimation error models.