Estimation of an unknown deterministic vector from quantized sensor data is considered in the presence of spoofing attacks which alter the data presented to several sensors. Contrary to previous work, a generalized attack model is employed which manipulates the data using transformations with arbitrary functional forms determined by some attack parameters whose values are unknown to the attacked system. For the first time, necessary and sufficient conditions are provided under which the transformations provide a guaranteed attack performance in terms of Cramer-Rao Bound (CRB) regardless of the processing the estimation system employs, thus defining a highly desirable attack. Interestingly, these conditions imply that, for any such attack when the attacked sensors can be perfectly identified by the estimation system, either the Fisher Information Matrix (FIM) for jointly estimating the desired and attack parameters is singular or that the attacked system is unable to improve the CRB for the desired vector parameter through this joint estimation even though the joint FIM is nonsingular. It is shown that it is always possible to construct such a highly desirable attack by properly employing a sufficiently large dimension attack vector parameter relative to the number of quantization levels employed, which was not observed previously. To illustrate the theory in a concrete way, we also provide some numerical results which corroborate that under the highly desirable attack, attacked data is not useful in reducing the CRB.Index Terms-Spoofing attack, distributed vector parameter estimation, Cramer-Rao Bound, the Expectation-Maximization algorithm, sensor network.
This paper considers the design of minimal mean square error (MMSE) transceivers in a wireless sensor network. The problem is nonconvex and challenging, and previous results (with partial solutions and/or with convergence unproved) left much to be desired. Here we propose several approaches-2 block coordinate descent (BCD), essentially cyclic multi-block method and its variants and distributive method to solve this problem. The proposed 2-BCD approach formulates the subproblem of joint beamformer optimization as a general second-order cone programming problem, which lends itself to standard numerical solvers and which requires no extra assumptions like previous works do. The proposed essentially cyclic multi-block approach further decomposes the joint beamformer design subproblem into multiple blocks, and rigorously solves each with semi-closed-form solution. The distributive algorithm optimizes transmitters in a decentralized manner and has never been considered in existing literature. The distributive algorithm has time complexity independent of number of sensors and is especially suitable for large-scale networks. All the previous BCD-based approaches left some singularity issue unattended as well as the convergence property unaddressed, our proposals are the first to provide a complete and provably-converging analytical solution. Extensive analysis and simulations demonstrate the merits of the novel approaches relative to existing alternatives.
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