Learning Requirements for Stealth Attacks

Abstract

The learning data requirements are analyzed for the construction of stealth attacks in state estimation. In particular, the training data set is used to compute a sample covariance matrix that results in a random matrix with a Wishart distribution. The ergodic attack performance is defined as the average attack performance obtained by taking the expectation with respect to the distribution of the training data set. The impact of the training data size on the ergodic attack performance is characterized by proposing an upper bound for the performance. Simulations on the IEEE 30-Bus test system show that the proposed bound is tight in practical settings.Index Termsstealth attacks, data injection attacks, random matrix theory, information theory

Asymptotic Learning Requirements for Stealth Attacks on Linearized State Estimation

“…Given the Wishart distribution of the attack covariance matrix in (18), the KL divergence objective in (21) and the cost functions in (22) are both random variables. Following on the same steps as in [18], we defined the ergodic performance of the attack as the performance obtained by averaging over all realizations of the training data set, i.e. as E[F (Σ Ã Ã)].…”

“…Under this scenario, only probabilistic bounds are available for the eigenvalues of random matrices. To that end, we can only provide upper and lower bounds on the non-asymptotic ergodic performance [18].…”

“…In this paper, we characterize the learning requirements for the information-theoretic stealth attacks proposed in [11] and [12] via asymptotic analysis tools from random matrix theory (RMT). To that end, we adopt a sample covariance matrix estimation framework as in [18]. In this setting, the attacker computes the sample covariance matrix from past realizations of the state variables and uses the estimate of the covariance matrix to construct the attack vector.…”

“…The nonasymptotic performance, i.e. the case with a finite number of realizations to compute the sample covariance matrix, is studied in [18] but the study using RMT tools in [18] only provides bounds on the expected value of the performance. In this paper, we focus on the asymptotic regime and characterize analytically the ergodic attack performance and its variance.…”

Data-Injection Attacks

“…The stealth attack construction proposed in the preceding text requires perfect knowledge of the covariance matrix of the state variables and the linearized Jacobian measurement matrix. In [23], the performance of the attack when the second-order statistics are not perfectly known by the attacker but the linearized Jacobian measurement matrix is known. Therein, the partial knowledge is modelled by assuming that the attacker has access to a sample covariance matrix of the state variables.…”