Elsevier

Automatica

Volume 127, May 2021, 109537
Automatica

Brief paper
Encryption scheduling for remote state estimation under an operation constraint

https://doi.org/10.1016/j.automatica.2021.109537Get rights and content

Abstract

In remote state estimation, data transmitted by a sensor through a wireless communication channel may be overheard by an eavesdropper. One possible way to avoid information leakage is to encrypt the transmitted data all the time. However, this may impose an extra operation energy burden on the sensor. In this paper, we investigate the optimal encryption scheduling in order to protect data privacy and ensure estimation accuracy under an energy constraint. Specifically, the sensor computes its local state estimate and then quantizes it using a non-subtractively dithered quantizer. Before each transmission, the sensor determines whether encrypting the data or not in order to strike a balance between data privacy and estimation accuracy. As the information about eavesdropper is unknown to the estimator, we introduce the concept of eavesdropper-invariant schedules and derive associated structural results. In addition, we propose a practical algorithm that compares a finite number of points to obtain an ε-optimal encryption schedule. Numerical examples are provided to illustrate performance benefits of the proposed methods.

Introduction

Cyber–physical systems (CPSs) which equip physical processes with sensing, computing and communication capabilities(Poovendran et al., 2012) have been widely applied in different realms, such as transportation system, smart grid, and wearable medical systems. However, the vulnerability of cyber layer exposes the physical processes to many potential threats and malicious attacks. For example, the transmitted data containing confidential information about the physical processes is broadcast over air and is easy to be overheard by neighboring devices, which may lead to severe economic losses and significantly threaten human lives if the data is used maliciously (Aysal & Barner, 2008). In this paper, we investigate a countermeasure for the privacy leakage for remote state estimation, a typical application scenario of CPSs. Specifically, a sensor forms a local estimate of the state of a dynamic system and transmits it to a remote estimator over a wireless channel, which is vulnerable to be intercepted by an eavesdropper.

Different from active attacks, e.g., denial-of-service (DoS) attacks (Li et al., 2017, Liu et al., 2014, Zhang et al., 2016) and deception attacks (Guo et al., 2017, Mo et al., 2014), eavesdropping is passive and violates the data confidentiality. Traditionally, one possible remedy to prevent information leakage (to eavesdroppers) is using cryptographic techniques, whereas the implementation is complicate in practice. Considering the wireless sensors in CPSs often have limited power (Ashibani & Mahmoud, 2017), the literature has proposed several alternative ways. One example is adding uncertainty to wiretap channels, which exploits the physical layer to achieve confidentiality (Wiese et al., 2016, Wiese et al., 2019). Even if the eavesdropper has unlimited computational capability, it is possible to ensure privacy with zero-error secrecy. However, such methods are difficult to implement in practice as the eavesdropper’s channel knowledge is unknown to the legitimate system.

Motivated by the difficulty of uncertain wiretap channels, a control-theoretic approach which follows a packet-based paradigm (Sinopoli et al., 2004) was utilized by Leong, Quevedo, Dolz, and Dey (2019) and Tsiamis et al., 2017a, Tsiamis et al., 2017b, where the estimator and the eavesdropper receive and intercept packets through a lossy channel with different arrival rates. The idea of perfect secrecy was first introduced by Tsiamis et al. (2017b). It requires any estimation error of the eavesdropper to be divergent and that of the legitimate user to be bounded. Under the condition that the legitimate receiver’s reception rate is larger than that of the eavesdropper, perfect secrecy can be achieved by proper transmission schedules. In Tsiamis et al. (2017a), the same authors illustrated that an essential event that the user receives the packet while the eavesdropper fails to intercept it, occurs once perfect secrecy has been achieved. Leong et al. (2019) further proposed a Markovian transmission schedule to achieve perfect secrecy even when the eavesdropper has a greater probability to obtain information.

Different from existing works, which treated transmitting nothing as a special encryption method, in this work we consider a more general cryptographic operation. To prevent the eavesdropper from obtaining information by intercepting the associated encrypted messages, the key for encryption is changed at each transmission. Similar to Leong et al. (2019) and Tsiamis et al., 2017a, Tsiamis et al., 2017b, in the problem setup we take into account the influence of decryption on estimation accuracy. Due to energy limitation at the sensor side, we aim to encrypt some of the transmitted packets, while leaving the others unencrypted. In short, we attempt to derive an optimal encryption schedule to minimize a weighted cost of estimation accuracy and data privacy under an energy constraint over an infinite-time horizon. Intermittent encryption in networked systems is an idea that has been studied in the Automatic Control community and, to the best of our knowledge, it is not currently utilized in practical or commercial applications. However, this study might be used as a guide in the future. In our previous related work (Huang, Leong, Quevedo, & Shi, 2019), we analyzed the scheduling problem in a finite-time horizon, in which a threshold-based policy was derived. As mentioned in Huang et al. (2019), the threshold value depended on the time index and the number of belief states grew exponentially with the time horizon. Thus the approach in Huang et al. (2019) cannot be extended straightforwardly to the infinite-time horizon case.

By comparison, the main contributions of this paper are summarized as follows:

  • (1)

    A general framework:The problem formulation in this work is more general since the encryption method can cover the methods proposed in Leong et al. (2019) and Tsiamis et al., 2017a, Tsiamis et al., 2017b. Moreover, we allow stochastic schedules, thus the action space is enlarged from a binary set to a set of probability distributions.

  • (2)

    A new algorithm: We focus on a practical schedule which does not rely on the knowledge of the eavesdropper’s error covariance, namely, the eavesdropper-invariant schedule (Definition 1). A primal decomposition is used to derive the optimal schedule: fix the average encryption rate, and then the equivalent master problem is proved to be piecewise concave in terms of the encryption rate (Theorem 4). Algorithm 1 is further proposed to derive a ε-optimal schedule by comparing a finite number of points (Lemma 5).

The remainder of this manuscript is organized as follows: In Section 2, we introduce the system model and provide a mathematical formulation of the problem. The main results are stated in Section 3. Section 4 presents numerical simulations and Section 5 draws conclusions.

Notation: N is the set of natural numbers. R and Rn represent the set of real numbers and ndimensional real column vectors. For a matrix X, X, tr(X) and ρ(X) denote its transpose, trace and spectral radius, respectively. When X is a positive semidefinite matrix, it is written as X0. E[] and E[|] are the expectation and conditional expectation. The notation P[] refers to probability. N(μ,Σ) denotes a Gaussian distribution with mean μ and covariance matrix Σ. For functions f, fk is defined as fk(X)=fffk times(X), with f0(X)=X.

Section snippets

System model

The system structure is shown in Fig. 1 with a LTI process: xk+1=Axk+wk,yk=Cxk+vk,where xkRn is the system state, ykRm is the measurement vector taken by the sensor at time k, wkRn and vkRm are two i.i.d. zero-mean Gaussian random noises with covariances Q0 and R>0, respectively. The initial state x0N(0,Π0) is uncorrelated with wk and vk. We further assume that (A, Q) is controllable and (A,C) is observable.

A “smart” sensor is equipped with a Kalman filter to compute its local state

Main results

In this section, we study the solution for the above optimization problem. First, we show the uniqueness of the limiting state distribution under a general stationary schedule. Furthermore, for any SEIE schedule, we obtain a closed-form of the limiting distributions of the remote estimator’s and the eavesdropper’s holding times. Since Je solely depends on the average encryption rate r, we adopt a primal decomposition to solve the problem and show that a threshold structure is necessary for an

Simulation

In this section, we consider a 2-dimensional system with parameters A=1.50.101, C=12, Q=0.5000.5and R=0.6, p1=0.9 and p2=0.8. We provide three examples to illustrate our results.

Conclusion

In this work, we studied an optimal EIE schedule to minimize the weighted cost of estimation accuracy and data privacy over an infinite-time horizon given a energy constraint. We showed the existence and threshold structure of an optimal SEIE schedule among EIE schedules under stability conditions. The ε-optimal SEIE schedule is further derived by comparing a finite number of values. We also discussed the unbounded scenarios. Extensions to other encryption methods and multiple sensors will be

Lingying Huang received her B.S. degree in Electrical Engineering and Automation from Southeast University, Nanjing, China, in 2017. She is currently a Ph.D. candidate at the School of Electrical and Electronic Engineering, Hong Kong University of Science and Technology. From July 2015 to August 2015, she had a summer program in Georgia Tech University, USA. Her current research interests include cyber–physical system security/privacy, networked state estimation, event-triggered mechanism and

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  • Cited by (14)

    • Transmission scheduling for privacy-optimal encryption against eavesdropping attacks on remote state estimation

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      Citation Excerpt :

      In this section, we first show that the above results in the FTH can be extended to the IPTH directly. Then, considering that the channel between the sensor and estimator is unreliable in actual industrial environment (Huang, Ding, Leong, Quevedo, & Shi, 2021; Qin et al., 2018; Wang et al., 2021), we design an algorithm to find suboptimal solution based on the monotonicity of constructed scheduling sequence in the following Proposition 2. Since the channel state information of the eavesdropper is difficult to obtain, we still assume that the eavesdropper will be able to obtain the information when the sensor sends plaintext packets.

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    Lingying Huang received her B.S. degree in Electrical Engineering and Automation from Southeast University, Nanjing, China, in 2017. She is currently a Ph.D. candidate at the School of Electrical and Electronic Engineering, Hong Kong University of Science and Technology. From July 2015 to August 2015, she had a summer program in Georgia Tech University, USA. Her current research interests include cyber–physical system security/privacy, networked state estimation, event-triggered mechanism and distributed optimization.

    Kemi Ding received her B.S. degree in Electronic and Information Engineering from Huazhong University of Science and Technology, Wuhan, China, in 2014 and the Ph.D. degree in the Department of Electronic and Computer Engineering from Hong Kong University of Science and Technology, Kowloon, Hong Kong, China, in 2018. She is currently a postdoctoral researcher at the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. Prior to this, she was a postdoctoral researcher in the School of Electrical, Computer and Energy Engineering, Arizona State University from September 2018 to August 2019. Her current research interests include cyber–physical system security/privacy, networked state estimation, game theory and graph signal processing.

    Alex Leong was born in Macau in 1980. He received the B.S. degree in mathematics and B.E. degree in electrical engineering in 2003, and the Ph.D. degree in electrical engineering in 2008, all from the University of Melbourne, Australia. He is currently with Defence Science and Technology Group, Australia. He was with the University of Melbourne from 2008 to 2015, and Paderborn University, Germany, from 2016 to 2019. His research interests include networked control systems, signal processing for sensor networks, and statistical signal processing. Dr. Leong was the recipient of the L. R. East Medal from Engineers Australia in 2003, an Australian Postdoctoral Fellowship from the Australian Research Council in 2009, and a Discovery Early Career Researcher Award from the Australian Research Council in 2012.

    Daniel E. Quevedo received Ingeniero Civil Electrónico and M.Sc. degrees from Universidad Técnica Federico Santa María, Valparaíso, Chile, in 2000, and in 2005 the Ph.D. degree from the University of Newcastle, Australia. He is Professor of Cyberphysical Systems at the School of Electrical Engineering and Robotics, Queensland University of Technology (QUT), in Australia. Before joining QUT, he established and led the Chair in Automatic Control at Paderborn University, Germany.

    Prof. Quevedo’s research interests are in networked control systems, control of power converters and cyberphysical systems security. He currently serves as Associate Editor for IEEE Control Systems and in the Editorial Board of the International Journal of Robust and Nonlinear Control. From he was Chair of the IEEE Control Systems Society Technical Committee on Networks & Communication Systems.

    In 2003 he received the IEEE Conference on Decision and Control Best Student Paper Award and was also a finalist in 2002. Prof. Quevedo is co-recipient of the 2018 IEEE Transactions on Automatic Control George S. Axelby Outstanding Paper Award. He is a Fellow of the IEEE.

    Ling Shi received the B.S. degree in electrical and electronic engineering from Hong Kong University of Science and Technology, Kowloon, Hong Kong, in 2002 and the Ph.D. degree in Control and Dynamical Systems from California Institute of Technology, Pasadena, CA, USA, in 2008. He is currently a Professor in the Department of Electronic and Computer Engineering, and the associate director of the Robotics Institute, both at the Hong Kong University of Science and Technology. His research interests include cyber–physical systems security, networked control systems, sensor scheduling, event-based state estimation and exoskeleton robots. He is a senior member of IEEE. He served as an editorial board member for the European Control Conference 2013–2016. He was a subject editor for International Journal of Robust and Nonlinear Control (2015–2017). He has been serving as an associate editor for IEEE Transactions on Control of Network Systems from July 2016, and an associate editor for IEEE Control Systems Letters from Feb 2017. He also served as an associate editor for a special issue on Secure Control of Cyber Physical Systems in the IEEE Transactions on Control of Network Systems in 2015–2017. He served as the General Chair of the 23rd International Symposium on Mathematical Theory of Networks and Systems (MTNS 2018). He is a member of the Young Scientists Class 2020 of the World Economic Forum (WEF).

    The work by L. Huang and L. Shi is supported by a Hong Kong RGC General Research Fund 16204218. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Juan C. Aguero under the direction of Editor Torsten Söderström.

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