Asymptotically Optimum Distributed Estimation in the Presence of Attacks

Distributed estimation of a deterministic mean-shift parameter in additive zero-mean noise is studied when using quantized data in the presence of Byzantine attacks. Several subsets of sensors are assumed to be tampered with by adversaries using different attacks such that the compromised sensors transmit fictitious data. First, we consider the task of identifying and categorizing the attacked sensors into different groups according to distinct types of attacks. It is shown that increasing the number K of time samples at each sensor and enlarging the size N of the sensor network can both ameliorate the identification and categorization, but to different extents. As K→∞, the attacked sensors can be perfectly identified and categorized, while with finite but sufficiently large K, as N→∞, it can be shown that the fusion center can also ascertain the number of attacks and obtain an approximate categorization with a sufficiently small percentage of sensors that are misclassified. Next, in order to improve the estimation performance by utilizing the attacked observations, we consider joint estimation of the statistical description of the attacks and the parameter to be estimated after the sensors have been well categorized. When using the same quantization approach successfully employed without attacks, it can be shown that the corresponding Fisher Information Matrix (FIM) is singular. To overcome this, a time-variant quantization approach is proposed, which will provide a nonsingular FIM, provided that K ≥ 2. Furthermore, the FIM is employed to provide necessary and sufficient conditions under which utilizing the compromised sensors in the proposed fashion will lead to better estimation performance when compared to approaches where the compromised sensors are ignored.