Remote State Estimation with Privacy Against Active Eavesdroppers

In recent years, Cyber-Physical Systems (CPS) have grown rapidly with the ubiquitous rise of the Internet of Things. CPS integrate cyber-infrastructure and physical systems. These systems encompass large-scale, geographically dispersed systems, such as smart grids, intelligent transportation systems, critical infrastructure and wearable medical devices [12]. Due to the tight integration of cyber and physical components in such systems, CPS are vulnerable to cyber-attacks. Cyber vulnerabilities can be exploited by malicious adversaries, which seek to disrupt CPS operations. Attacks on CPS may result in performance degradation or even system failure [11].

Security of a system is defined by availability, integrity, and confidentiality. Targeting these security goals are; denial-of-service (DoS), deception attacks, and eavesdropping. DoS attacks seek to compromise the availability of information via jamming communication channels. Deception attacks seek to target the integrity of a system by manipulating transmitted information. Eavesdropping attacks seek to intercept transmitted information, targeting confidentiality, [4]. In particular, the use of shared wireless mediums makes it difficult to shield transmissions from unintended recipients. Sophisticated attacks may combine elements to achieve goals, such as active eavesdropping. Active eavesdropping is where the eavesdropper can not only intercept transmissions but also invade the communication channel to degrade the wireless performance of the system [7]. This performance degradation may trigger the sensor to transmit data packets more frequently, or with higher power which increases the packet arrival rate at the eavesdropper. Therefore, estimation accuracy at the eavesdropper is increased. This style of attack differs from a DoS attack, as jamming is utilised to improve the eavesdropper performance as well, whereas a DoS attack seeks to degrade the legitimate user’s performance, with no consideration to its own packet reception. In practical applications, active eavesdropping is achieved through pilot signal contamination [24].

There is a need to design estimation algorithms over wireless communication networks that account for the reliability of the wireless network, while also ensuring confidentiality of the transmitted information from an eavesdropper. Guaranteeing confidentiality is crucial for sensor and control data, which convey confidential information about the physical system state. This is detrimental to system operation as it violates sensor and user privacy, which may cause economic losses or even threaten human survival. For example, in smart grids an eavesdropper can infer load taxonomy from the electricity consumption data provided by smart meters. The malicious adversary can infer daily schedules, and break into premises when no inhabitants are present [20]. In addition, with knowledge of how a system operates, an adversary may perform an attack with greater sophistication.

To ensure privacy against eavesdropping attacks, cryptography-based tools [13] are at times used in practice, however, they introduce computation and communication overheads [15] and therefore are only of limited use. Several works have developed encoding schemes [17, 22] and scheduling of transmissions [16], which are lightweight and avoid significant computation. These encoding techniques require knowledge of the legitimate estimator’s performance. Performance is shared through acknowledgment of successful packet receipts to the sensor. However, utilising an acknowledgement channel is a point of vulnerability for eavesdropping and DoS attacks [7], as experimentally demonstrated with the jamming of acknowledgments utilising commodity hardware [14]. Therefore, the motivation for this work is to design a low-complexity encoding scheme in a system without acknowledgments when an eavesdropper attack is occurring, by adapting transmissions to ensure the confidentiality of the state.

In this work, we consider a sensor transmitting the state estimate of a stable, linear, dynamic system to a legitimate remote user over a random packet-dropping wireless network. A malicious adversary may perform an active eavesdropping attack during system operation and intercept the transmissions from the sensor, compromising state confidentiality. We propose an encoding scheme that is activated on the detection of an eavesdropper. We present a Quickest Change Detection (QCD) scheme to detect the presence of an eavesdropper through monitoring channel outcomes. QCD is a technique to detect an abrupt change in statistical properties of an observed process as quickly as possible after the change occurs, subject to a false alarm constraint [2]. Several formulations for QCD problems exist, which vary with respect to assumptions about the change time and optimality criteria. Bayesian formulations assume the change time is a random variable with a known geometric prior with independent and identically distributed (i.i.d.) observations [10, 23]. With these assumptions, Shiryaev established the optimal stopping, which compared the change posterior probability to a threshold.

In this work, we use QCD to detect the presence of an active eavesdropper, which activates the encoding scheme. When an eavesdropper is detected, the sensor transmits noise or its local state estimate, based on a pseudo-random Bernoulli indicator that is pre-arranged at the legitimate user and sensor. Under this encoding scheme, the eavesdropper’s estimation error covariance is made larger than that of the legitimate user. This work extends the results in [5]. We extend [5] from the scalar, first-order system to the vector case. Additionally, no method was presented in [5] for the detection of the eavesdropper.

We consider the following dynamic system that is modelled as a linear, discrete-time invariant system

where $x_{k}\in\mathbb{R}^{n}$ is the system state vector and $y_{k}\in\mathbb{R}^{m}$ is the measurement taken by the sensor at time $k\geq 0$. The dynamics is stable with $\rho(A)<1$, where $\rho(.)$ is the spectral radius¹¹1The spectral radius is defined as the maximum absolute eigenvalue $\rho(.)=\max|\beta_{i}(.)|$ where $\beta_{i}$ is the $i$th eigenvalue.. The process noise $w_{k}\in\mathbb{R}^{n}$ and the measurement noise $v_{k}\in\mathbb{R}^{m}$ are zero mean i.i.d. Gaussian random processes with $\mathbb{E}[w_{k}w_{k}^{\intercal}]=Q$ with $\mathbb{E}[w_{k}w_{l}^{\intercal}]=0$ and $\mathbb{E}[v_{k}v_{k}^{\intercal}]=R$ with $\mathbb{E}[v_{k}v_{l}^{\intercal}]=0$ and $\mathbb{E}[w_{k}v_{l}^{\intercal}]=0\ \text{ for all }l\geq 0\text{ \ where\ }k\neq l$. The initial state of the process $x_{0}$ is a Gaussian random variable with zero mean and covariance $\mathbb{E}[x_{0}x_{0}^{\intercal}]=\Sigma_{0}$. The covariances $Q,R,\Sigma_{0}$ are positive definite. The initial state is uncorrelated with $w_{k}$ and $v_{k}$ with $\mathbb{E}[x_{0}w_{k}^{\intercal}]=0$ and $\mathbb{E}[x_{0}v_{k}^{\intercal}]=0$. All system and noise parameters $(A,C,Q,R,\Sigma_{0})$ are assumed to be public knowledge, available to the sensor, legitimate user and eavesdropper. The pair $(A,C)$ is assumed to be observable and $(A,\sqrt{Q})$ is controllable.

As packet receipt acknowledgments may be vulnerable to adversaries, our objective is to design an encoding scheme that is activated on the detection of an eavesdropper, that does not rely on acknowledgements when the eavesdropper is detected. To characterise performance bounds under the encoding scheme for the eavesdropper’s and legitimate user’s estimation error covariance, we introduce “open-loop performance”. We first define the solution to the discrete-time Lyapunov equation, which is expressed as an infinite sum in the case that $\rho(T)<1$

where $V$ is the unique stabilising solution to

For convenience, we define the following notation to represent the solution to the Lyapunov equation

If no information is received from the sensor at the legitimate user or eavesdropper, then open loop performance occurs. It is characterised by $x^{OP}_{k}=\mathbb{E}[x_{k}]=0$ and $P^{OP}_{k}=\mathbb{E}[x_{k}x_{k}^{\intercal}]$, where the asymptotic open loop estimate and corresponding estimation error covariance satisfy [1]

The encoding scheme is designed with no information of the legitimate user’s current estimate, hence we aim to ensure that the trace of the legitimate user’s expected estimation error covariance is upper bounded by the trace of open loop prediction. To ensure confidentiality, we desire that the trace of the eavesdropper’s expected estimation error covariance is above the trace of open loop prediction, i.e., the trace of the expected eavesdropper’s estimation error covariance when using received packets is larger than if it had not used any packets in its estimation process. These conditions guarantee data confidentiality against an eavesdropper, whilst also ensuring the legitimate user has a reliable estimate.

The considered remote estimation architecture is shown in Fig. 1 and consists of a sensor observing a dynamic system, a legitimate user, and an eavesdropper. The sensor transmits packets of information over a wireless channel. To degrade the eavesdropper’s estimation performance, we desire to limit the amount of useful information it receives. To actively harm its estimation performance, we propose to send noise intermittently.

In an unreliable, insecure network, the sensor’s state estimate may not be received by the legitimate user, or the received packet may be noise due to the employed encoding scheme. To estimate the state of the system with intermittent observations, a second estimation scheme is employed at the legitimate user.

The legitimate user obtains the state estimate as follows: Once the sensor’s transmission arrives, and the sensor has not transmitted noise, the estimator synchronizes $\hat{x}_{k}$ with the optimal state estimate of the sensor $\hat{x}_{k}^{s}$. Otherwise, the legitimate user predicts $x_{k}$ based on the previous estimate using the system model in (1). Thus, the state estimate at the legitimate user $\hat{x}_{k}$ (12) satisfies

The corresponding estimation error covariance $P_{k}$ satisfies

It follows from the eavesdropper’s estimation error covariance in (14) that when a packet arrives and it is the state estimate from the sensor, the legitimate user’s estimation error covariance is that of the sensor’s $\bar{P}$ (see (2)). When a packet of noise arrives, or a packet dropout occurs, the legitimate user’s estimation error covariance grows.

The goal of the eavesdropper is to construct an estimate of the state of the system $x_{k}$ based on all sensor data packets received up to time step $k$. The eavesdropper is unaware of the detection variable $\nu_{k}$ (9). The eavesdropper is additionally unaware of the randomised indicator to send noise $u_{k}$ (10). As the noise is designed to exhibit the same statistical characteristics as the sensor’s state estimate, and the encoding scheme is unknown, the eavesdropper cannot distinguish $\hat{x}_{k}^{s}$ from $n_{k}$. This feature can be used to deceive the eavesdropper to incorporate noise $n_{k}$ into its state estimate. The eavesdropper obtains the state estimate as follows: Once the sensor’s transmission arrives, and the sensor has not transmitted noise, the eavesdropper synchronizes $\hat{x}^{e}_{k}$ with that of the sensor. If the sensor’s transmission arrives, and the sensor has transmitted noise, the eavesdropper synchronizes $\hat{x}^{e}_{k}$ with the transmitted noise $n_{k}$. If a packet dropout occurs, the eavesdropper predicts $x_{k}$ based on the previous estimate using the system model in (1), such that the eavesdropper’s state estimate $\hat{x}_{k}^{e}$ (13) from the sensor’s viewpoint satisfies

We characterise the eavesdropper’s estimation error covariance in Lemma 1.

Lemma 1 characterises the instantaneous worst-case performance for the eavesdropper under the encoding scheme. It shows that if noise is transmitted often, we can drive the eavesdropper’s estimation error covariance above open-loop prediction. However, greater transmissions of noise degrade the legitimate user’s performance as well. Therefore, the probability of the sensor sending noise $\mu$ needs to be chosen to balance the impact between degrading the legitimate user’s performance and protecting data confidentiality.

To quantify the estimation performance of the legitimate user and eavesdropper under the encoding scheme from a sensor viewpoint, we explore the mathematical expectation of the estimation error covariance. The use of the expectation allows us to compute the expected performance over the stochastic channel and encoding scheme. Particularly, due to the absence of acknowledgments when the eavesdropper is performing an attack, the channel outcomes are unknown at the sensor. Thus we use the expected estimation performance to design the sensor’s encoding variable $\mu$. We additionally explore the steady-state behaviour of the expected estimation error covariance to quantify long-term performance under the encoding scheme as $k\rightarrow\infty$.

We present an analytic expression of the steady state expectation of estimation error covariance of the legitimate user in Lemma 2 and of the eavesdropper in Lemma 3. Our expressions depend on the channel qualities (3), (4), process dynamics (1), and the encoding design variable (11).

The proof is included in Section 6.

The proof is included in Section 6.

The mathematical expectation of the legitimate user’s and eavesdropper’s estimation error covariance at time $k$ can be found by taking the sum of all possible channel and sensor transmission outcomes multiplied by the corresponding probability of that channel and sensor transmission realisation occurring. To find the mathematical expectation of the eavesdropper’s and legitimate user’s estimation error covariance, we introduce the following Markov chain models.

To provide a range of the encoding design variable $\mu$ (11), we write the expectations of the estimation error covariances as a function of the encoding design variable. The expectation of the steady-state estimation error covariance of the legitimate user from Lemma 2 can be written as

where $W$ and $S$ are functions of the encoding design variable $\mu$. Similarly, the expectation of the steady-state estimation error covariance of the eavesdropper from Lemma 3 can be written as

Before providing a method to select the encoding design variable $\mu$, we observe the following monotonicity result.

Consider $0<\mu^{\star}<\mu<1$. From the trace of the eavesdropper’s expected steady-state estimation error covariance (see (21)), the inequality $J^{e}(\mu^{\star})<J^{e}(\mu)$ is written as

By algebraic manipulation

From [19, Lemma 1], let $P^{*}=W^{e}$, $\hat{P}=H^{e}$ with $Q^{*}=\bar{P}$ and $\hat{Q}=P_{n}$. As $P_{n}=P^{OP}+\bar{P}$, it shows that $P_{n}>\bar{P}$. Therefore $H^{e}>W^{e}$. As $\operatorname{trace}H^{e}>\operatorname{trace}W^{e}$, it shows that $\operatorname{trace}H^{e}-\operatorname{trace}W^{e}>0$. Therefore from the inequality (23), it is shown that

which completes the proof. Considering the choice of $\mu$ in our encoding scheme, we desire to meet two conditions. Firstly, the trace of the legitimate user’s expected steady-state estimation error covariance should be upper bounded by that of the trace of open-loop prediction, with $J(\mu)<\operatorname{trace}P^{OP}$. This condition ensures that transmissions from the sensor remain informative to the legitimate user.

Secondly, the trace of the eavesdropper’s expected steady-state estimation error covariance should be greater than the trace of open loop prediction, with $J^{e}(\mu)>\operatorname{trace}P^{OP}$. This condition ensures that the confidentiality of the state is protected. The following Lemma provides an upper bound for the trace of the legitimate user’s steady-state estimation error covariance.

Consider the encoding design variable $\mu<1$ and legitimate user’s channel quality $\bar{\gamma}<1$, and the eavesdropper has been detected with $\nu_{k}=1$. Suppose that there is a non-zero probability that the sensor’s state estimate is received at the legitimate user $\mathbb{P}[\lambda_{k}=1,\nu_{k}=1,u_{k}=0]>0$. Then, from the Borel–Cantelli Lemma [8], a packet containing the state estimate from the sensor will arrive at the legitimate user with probability one for some time $k>0$, that is, $\mathbb{P}[\lambda_{k}=1,\nu_{k}=1,u_{k}=0,\text{ for some }k>0]=1$. The estimation error covariance of the legitimate user when the state estimate is received is $\bar{P}$, noting that $\bar{P}<P^{OP}$. Therefore, $\lim_{k\to\infty}\mathbb{E}[P_{k}]<P^{OP}$ if $\mu<1$ and $\bar{\gamma}<1$ which shows $J(\mu)<\operatorname{trace}P^{OP}$. This completes the proof. The following theorem provides a range for $\mu$ that protects data confidentiality, whilst keeping transmissions informative to the legitimate user.

In comparison with the state secrecy codes developed for stable systems [22], the current method allows one to drive the eavesdropper’s estimation error covariance above open-loop prediction through transmitting noise with $\mu>\mu^{OP}$. Thus, interceptions of packets are not informative to the eavesdropper’s estimation process. Moreover, the eavesdropper intercepting packets is actively harming its own estimation performance. With the eavesdropper’s expected estimation error covariance above open-loop prediction, the eavesdropper’s performance is worse than if it did not utilise any packets at all from the sensor in its estimation process, i.e., if the eavesdropper only predicted the state estimate using the system model in (1). Since sending noise harms the legitimate user’s performance as well, we choose $\mu<1$ so the performance of the legitimate user remains upper-bounded by open loop prediction. This ensures that transmitted packets from the sensor remain informative to the legitimate user for use in estimation.

In this section, we illustrate the performance of the encoding scheme using some examples. Consider the following second-order system (1) with parameters $A=\begin{bmatrix}0.5&0.1\\ 0.4&0.6\end{bmatrix}$, $C=I_{2}$, $Q=R=10^{-2}I_{2}$, where $I$ is the identity, and $\rho(A)=0.7562<1$. This gives $\operatorname{trace}\bar{P}=0.0114$, $\operatorname{trace}P_{n}=0.0502$ and $\operatorname{trace}P^{OP}=0.0388$. Consider a legitimate user channel quality of $\bar{\gamma}=0.5$ and four cases of eavesdropper channel qualities, with $\gamma^{e}_{1}=0.7$, $\gamma^{e}_{2}=\bar{\gamma}$ $\gamma^{e}_{3}=0.3$. We also consider the case that $\gamma^{e}=0,\bar{\gamma}=1$. Fig. 4 shows the difference of the traces of the steady-state expected estimation error covariance between the eavesdropper and legitimate user $(J^{e}(\mu)-J(\mu))/J(\mu)$ against the encoding design variable $\mu$ for these four cases.

In the case that the eavesdropper’s channel is worse than the legitimate user’s, with $\bar{\gamma}<\gamma^{e}_{1}$ then the trace of the eavesdropper’s estimation covariance is greater than the legitimate user’s for $\mu\geq 0$ with $(J^{e}(\mu)-J(\mu))/J(\mu)>0$. Using Theorem 6, $\mu$ is chosen in the range $0.7178<\mu<1$. In the case that the eavesdropper’s channel is equal to the legitimate user’s with $\bar{\gamma}=\gamma^{e}_{2}$, then the trace of the eavesdropper’s estimation covariance is greater than the legitimate user’s for $\mu>0$, with $(J^{e}(\mu)-J(\mu))/J(\mu)>0$ if $\mu>0$, and $(J^{e}(\mu)-J(\mu))/J(\mu)=0$ if $\mu=0$. Using Theorem 6, $\mu$ is chosen in the range $0.7125<\mu<1$. In the case that the legitimate user’s channel is worse than the eavesdropper’s with $\gamma^{e}_{3}<\bar{\gamma}$, then the trace of the eavesdropper’s estimation covariance is greater than the legitimate user’s for $\mu>0.14$. For $\mu\leq 0.14$, the trace of the legitimate user’s expected estimation error covariance is greater than that of the eavesdropper’s. Using Theorem 6, $\mu$ is chosen in the range $0.7074<\mu<1$.