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In this paper we study the effect of false data injection attacks on state estimation carried over a sensor network monitoring a discrete-time linear time-invariant Gaussian system. The steady state Kalman filter is used to perform state estimation while a failure detector is employed to detect anomalies in the system. An attacker wishes to compromise the integrity of the state estimator by hijacking a subset of sensors and sending altered readings. In order to inject fake sensor measurements without being detected the attacker will need to carefully design his actions to fool the estimator as abnormal sensor measurements would result in an alarm. It is important for a designer to determine the set of all the estimation biases that an attacker can inject into the system without being detected, providing a quantitative measure of the resilience of the system to such attacks. To this end, we will provide an ellipsoidal algorithm to compute its inner and outer approximations of such set. A numerical example is presented to further illustrate the effect of false data injection attack on state estimation.