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Coding Schemes for Securing Cyber-Physical Systems Against Stealthy Data Injection Attacks

Fei Miao, Quanyan Zhu, Miroslav Pajic, George J. Pappas

arXiv:1605.08962v1cs.CReess.SY

TL;DR

Stealthy false-data attacks can exploit system-model knowledge to increase estimation errors while evading statistical detection. The paper uses low-cost sensor-output coding, computes feasible and multiple coding matrices, and introduces time-varying updates for attackers estimating the coding scheme. Its conclusion states that the approach addresses these attacks under the stated coding assumptions, while existing active monitoring has a formal limitation.

  • Problem

    Intelligent attackers can design sensor and actuator injections that increase state-estimation errors while passing the estimator and statistical detector.

  • Method

    The paper codes sensor outputs using feasible coding matrices, computes them algorithmically, and uses time-varying matrices with heuristic update timing when attackers estimate the scheme.

  • Results

    The paper derives conditions for feasible coding schemes, shows that multiple feasible matrices generally exist, and reports validity when attackers estimate the matrix through sensor and actuator measurements.

  • Takeaways & Limitations

    Coding sensor outputs provides a low-cost approach for detecting stealthy data injection attacks without requiring an additional detector.

  • Takeaways & Limitations

    For a Kalman-filtered system with a χ2 detector and appropriately functioning actuators, no active monitor of the specified form increases detection probability for stealthy sensor injections.

Abstract

from arXiv · show

This paper considers a method of coding the sensor outputs in order to detect stealthy false data injection attacks. An intelligent attacker can design a sequence of data injection to sensors and actuators that pass the state estimator and statistical fault detector, based on knowledge of the system parameters. To stay undetected, the injected data should increase the state estimation errors while keep the estimation residues small. We employ a coding matrix to change the original sensor outputs to increase the estimation residues under intelligent data injection attacks. This is a low cost method compared with encryption schemes over all sensor measurements in communication networks. We show the conditions of a feasible coding matrix under the assumption that the attacker does not have knowledge of the exact coding matrix. An algorithm is developed to compute a feasible coding matrix, and, we show that in general, multiple feasible coding matrices exist. To defend against attackers who estimates the coding matrix via sensor and actuator measurements, time-varying coding matrices are designed according to the detection requirements. A heuristic algorithm to decide the time length of updating a coding matrix is then proposed.

I. INTRODUCTION

Cyber-physical systems require security mechanisms that address control-layer challenges, including stealthy attacks that exploit system-model knowledge. This paper proposes low-cost sensor-output coding to expose such attacks without encryption across all measurements.

  • Cyber-physical systems integrate computation and communications with physical processes, making security critical in applications such as power, water, and safety infrastructure.
  • Intelligent attackers can increase state-estimation error while keeping detector residues small when they know the system model.
  • The paper addresses this threat with sensor-output coding that changes transmitted measurement values rather than correcting bit-level errors.
  • The coding method is presented as a low-cost alternative to encrypting all sensor measurements and requires no additional detector.
  • The paper develops conditions and an algorithm for feasible coding matrices, while considering estimator, detector, controller, sensor, and actuator components.
  • The linear system model uses state, control-input, and sensor-observation vectors with independent Gaussian process and measurement noises.

B. False data injection attack model

The attack model assumes an adversary can inject sensor and actuator data using knowledge of the system model. Stealth requires keeping residues close to normal behavior while driving estimation error, and feasibility depends on an unstable system mode aligned with attack controllability.

  • The attacker injects arbitrary vectors into sensor outputs and actuator inputs while knowing the system model and accessing the communication network.
  • With stable A−KCA and zero expected noises, the normal estimation error converges to zero while residues remain unlikely to trigger the alarm.
  • The attack-induced differences in residues and estimation errors are functions of the sensor and actuator injection sequences.
  • The attacker seeks to increase estimation error without triggering alarms and may aim to destabilize the system through unbounded estimation error.
  • Stealth is approximated by keeping compromised residues small and their change from normal residues bounded by an attacker-selected threshold M.
  • A stealthy injection exists if and only if A has an unstable eigenvalue whose eigenvector lies in the controllability subspace associated with (A−KCA, K).

III. CODING SENSOR OUTPUTS FOR DETECTING STEALTH SENSOR DATA INJECTION

The section shows that active monitors and observer-based fault filters cannot reliably detect intelligent stealthy sensor injections in the considered LTI setting. It motivates coding sensor outputs as an inexpensive alternative that increases detectability without relying on these limited approaches.

  • Limitations of existing approaches: Existing statistical detectors and fault-detection filters may miss stealthy sensor injections, even when actuators are uncompromised.Attackers can induce unbounded estimation error without detection under monitoring systems such as a χ2 detector.
  • Motivation: The section motivates inexpensive detection techniques to compensate for system vulnerability under intelligent sensor data injection attacks.The proposed direction avoids relying solely on active monitors or fault-detection filters.
  • Limitations of existing approaches: Active monitors that add linear control inputs cannot increase the detection probability of stealthy sensor data injections for a Kalman filter with a χ2 detector.Lemma 1 establishes this impossibility for active monitors of the specified form.
  • Limitations of existing approaches: The added control input is eliminated when comparing compromised and normal systems, so it does not increase the relevant residual difference.The proof states that the active input does not increase the norm of the residual difference under the attacked system.
  • Limitations of existing approaches: Different linear controllers are equivalent under stealth sensor data injection attacks in the unified LTI model.Consequently, the detection design does not restrict the controller model.
  • Limitations of existing approaches: Observer-based fault detectors share the Kalman filter’s limitation because their residual remains an observer-based difference between measured and estimated outputs.The residual signal is generated from the discrepancy between y_k and its estimate, leaving the intelligent injection stealthy.

B. Coding sensor outputs to detect stealth data injection

The paper codes sensor outputs with an invertible matrix Σ so attacks designed for the original system produce unbounded coded residuals under stated conditions. The scheme preserves the original estimator and relies on keeping Σ unknown to the attacker.

  • Coding sensor outputs: The coding scheme changes transmitted sensor outputs so stealthy injections designed for the original system increase coded estimation residues.The estimator and χ2 detector continue operating on decoded sensor outputs.
  • Detection effect: Under these conditions, the coded residue norm ∥∆z′_k∥2 increases to infinity as k →∞.This result applies when the attacker designs the injection using the original system parameters without knowing Σ.
  • Feasible coding matrices: A feasible coding matrix Σ is invertible and changes the direction of Cv for unstable eigenvectors v relevant to stealthy injections.Theorem 1 assumes (A, C) is detectable and v belongs to the specified controllability subspace.
  • System impact: The coding does not alter the physical matrices A and B, and the estimator still converges to the true state without attacks.The paper presents sensor-output transformation as the mechanism for obtaining different residues without changing the physical structure.
  • Multiple unstable eigenvectors: The construction extends to multiple unstable eigenvectors when Σ changes the direction of every relevant linear combination of their sensor images.Lemma 2 requires the coding matrix to satisfy the corresponding condition for any linear combination of the unstable eigenvectors.
  • Attacker knowledge: If the attacker learns Σ from sensor and actuator measurements, the system can send a new coding matrix before the current one is identified.The paper treats this update process as a response to an attacker estimating the coding scheme.

C. When sensor and actuator packets are both injected

The paper also considers attacks that inject both sensor and actuator packets. A suitable invertible sensor-output coding matrix can make the coded residual grow unbounded while the original attack remains stealthy.

  • Joint packet attacks: The analysis extends feasible coding to attacks that inject both sensor and actuator data packets.The actuator data is not coded and remains the same in the original and coded systems.
  • Detection condition: If the joint attack keeps the original residual bounded while driving estimation error to infinity, an appropriate invertible Σ makes the coded residual norm increase to infinity.Theorem 2 states this outcome after coded sensor values are injected and decoded by the estimator.
  • Transformed attack representation: The coded-system analysis represents the attack through the transformed sensor injection Σ−1ya_k while retaining the original Kalman-filter design.The actuator injection remains uncoded in the stated setup.

IV. ALGORITHM TO COMPUTE A CODING MATRIX

The paper constructs feasible coding matrices using Givens rotations applied to sensor space. The algorithm targets directions associated with unstable eigenvectors and provides an efficient, generally sparse transform under stated dimensional conditions.

  • Givens rotations: A Givens rotation changes only two coordinates by rotating a vector in a selected coordinate plane.The rotation uses c = cos θ and s = sin θ.
  • Coverage of attack directions: The rotation design aims to change every nonzero direction in span(Cv1, . . . , Cvu) so stealthy injections based on unstable eigenvectors are affected.The algorithm chooses rotations across the relevant sensor-coordinate space without requiring the injected data values.
  • Algorithm design: Algorithm 1 computes a feasible coding matrix from A, C, and the unstable eigenvalues and eigenvectors.It forms the sensor-space vectors Cvi, constructs a basis for their support, and composes selected rotations.
  • Existence guarantee: For sensor dimension p ≥2, a feasible Givens rotation matrix always exists under the theorem or lemma conditions.The proof uses rotation angles in (0, π/2] and avoids repeating the same rotation plane.
  • Computational cost: The coding scheme requires O(n^3+p^3) multiplications and additions, where n is the number of plant states and p is the number of sensors.The paper characterizes this as lower-cost than basic encryption and coding schemes using complex nonlinear primitives.
  • Efficiency and scope: The rotation matrix is generally sparse, and the algorithm is polynomial and heuristic, supporting computationally efficient coding.The paper identifies structural constraints and distributed coding as directions for future work.

V. TIME-VARYING CODING SCHEME WHEN THE ATTACKER ESTIMATES THE CODING MATRIX

The time-varying scheme addresses attackers who can estimate the coding design from observations. The matrix is updated according to attacker learning ability and system detection requirements.

  • Initial secrecy assumption: The coding scheme assumes the attacker initially does not know when Σ begins transforming sensor outputs or the exact coding matrix.The scheme is effective when sensor values are not manipulated before coding.
  • Time-varying defense: If the attacker learns the system model and coding design, the system should continually apply time-varying coding matrices.The update interval depends on the attacker’s learning ability and the system’s detection requirements.
  • Matrix updates: Each update costs the attacker time to infer the transformed sensor outputs, while sensors and the controller can synchronously regenerate or switch matrices.The paper assumes synchronized operation and secrecy of the coding matrix during communication.

A. The time length an attacker needs to learn Σ

The attacker estimates the coding matrix and initial state from eavesdropped sensor and actuator measurements, forming a bilinear estimation problem. Noise, non-convexity, and multiple solutions complicate recovery of the true coding matrix.

  • The attacker eavesdrops sensor outputs and actuator inputs to estimate the coding matrix instead of directly capturing it.
  • The observed measurements form bilinear equations involving the coding matrix Σ and the initial state x0.
  • N ≥ max{n, p} − 1 measurements are required to calculate the exact coding matrix Σ and true initial state x0 in the noise-free case.
  • With Gaussian noise, the attacker minimizes equation-fitting error because an exact solution for the true coding matrix is difficult to find numerically.
  • The optimization has a non-convex rank constraint, so the attacker uses a heuristic that initially ignores the constraint and checks full rank iteratively.
  • Different measurement durations can produce different optimal coding matrices, and multiple solutions may persist even after 20 measurements.

B. When the estimated ˆΣ̸ = Σ

An attacker using an estimated coding matrix faces a trade-off between collecting measurements and launching an injection. Time-varying coding limits the period during which the estimated matrix can support stealth.

  • An attacker who treats an estimated matrix ˆΣ as true cannot guarantee stealth because the residue depends on both Σ and ˆΣ.
  • When ˆΣ ≠ Σ, the estimated-matrix injection can remain stealthy for a longer time than the original-system injection.
  • Coding can extend the time before detection but cannot make the injection pass the detector, so update timing must balance attacker learning and stealth duration.
  • Longer measurement time can improve ˆΣ but may allow the system to trigger an alarm before the attacker recovers the coding scheme.
  • Insufficient measurements yield a poor estimate, so the resulting injection remains detectable quickly.
  • The system should update its coding matrix before the attacker’s estimated time proportion reaches the threshold ˜α(NΣ).
  • The heuristic chooses NΣ by repeatedly estimating ˆΣ from new measurements and increasing the measurement duration until α(NΣ) reaches ˜α(NΣ).

A. Coding scheme detect stealthy data injection

Two-dimensional linear-system examples show that coding can make attack residues grow while the original system’s residues remain bounded. A concrete threshold example shows detection can precede coding-matrix estimation.

  • The examples use a detectable two-dimensional linear system with an unstable eigenvalue λ = 1.
  • Multiple feasible coding matrices satisfying Theorem 1 exist for the example system, including Σ1.
  • Under coded outputs transformed by Σ1 and Σ2, residue changes increase over time, while the original system’s residue remains bounded.
  • The coded-system estimation error does not necessarily increase faster than the original system’s estimation error.
  • With residue threshold M = 2, an attack designed for the original system is detected after 12 seconds when applied to the coded system.

C. Number of measurements to estimate the coding matrix

Increasing the number of measurements improves the attacker’s estimate of the coding matrix, but does not necessarily extend stealth. In the reported case, N = 25 and N = 200 produce nearly identical residue-change norms, while the system can update its coding matrix every 5 seconds under the stated threshold.

  • C. Number of measurements to estimate the coding matrix: As N increases, the estimated coding matrix approaches the true coding matrix, while the residue-change norm increases more slowly under the corresponding injection sequence.This relationship is reported for the attacker’s estimate obtained from different numbers of measurements.
  • C. Number of measurements to estimate the coding matrix: For N = 25 and N = 200, the residue-change norms are almost identical, so more measurements do not yield a better stealth-preserving coding-matrix estimate in this case.The comparison is based on Figure 7’s two measurement counts.
  • C. Number of measurements to estimate the coding matrix: With threshold α̃(NΣ) = 1.5, heuristic Algorithm 3 selects a 5-second coding-matrix update interval.The interval follows from the reported stealth-duration comparison and threshold setting.
  • C. Number of measurements to estimate the coding matrix: The proposed coding scheme is intended to detect adaptive injections designed from an estimated coding matrix without requiring knowledge of the coding matrix applied by the system.The conclusion states this validity for attackers estimating the coding matrix through sensor and actuator measurements.
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