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IEEE/CAA Journal of Automatica Sinica

Quantization and Event-Triggered Policy Design for Encrypted Networked Control

Yongxia Shi, Ehsan Nekouei

Published Mar 18, 2024• 16 citations
Publisher

Abstract

This paper proposes a novel event-driven encrypted control framework for linear networked control systems (NCSs), which relies on two modified uniform quantization policies, the Paillier cryptosystem, and an event-triggered strategy. Due to the fact that only integers can work in the Pailler cryptosystem, both the real-valued control gain and system state need to be first quantized before encryption. This is dramatically different from the existing quantized control methods, where only the quantization of a single value, e.g., the control input or the system state, is considered. To handle this issue, static and dynamic quantization policies are presented, which achieve the desired integer conversions and guarantee asymptotic convergence of the quantized system state to the equilibrium. Then, the quantized system state is encrypted and sent to the controller when the triggering condition, specified by a state-based event-triggered strategy, is satisfied. By doing so, not only the security and confidentiality of data transmitted over the communication network are protected, but also the ciphertext expansion phenomenon can be relieved. Additionally, by tactfully designing the quantization sensitivities and triggering error, the proposed event-driven encrypted control framework ensures the asymptotic stability of the overall closed-loop system. Finally, a simulation example of the secure motion control for an inverted pendulum cart system is presented to evaluate the effectiveness of the theoretical results.

DOI
10.1109/jas.2023.124101

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This paper proposes a novel event-driven encrypted control framework for linear networked control systems (NCSs), which relies on two modified uniform quantization policies, the Paillier cryptosystem, and an event-triggered strategy. Due to the fact that only integers can work in the Pailler cryptosystem, both the real-valued control gain and system state need to be first quantized before encryption. This is dramatically different from the existing quantized control methods, where only the quantization of a single value, e.g., the control input or the system state, is considered. To handle this issue, static and dynamic quantization policies are presented, which achieve the desired integer conversions and guarantee asymptotic convergence of the quantized system state to the equilibrium. Then, the quantized system state is encrypted and sent to the controller when the triggering condition, specified by a state-based event-triggered strategy, is satisfied. By doing so, not only the security and confidentiality of data transmitted over the communication network are protected, but also the ciphertext expansion phenomenon can be relieved. Additionally, by tactfully designing the quantization sensitivities and triggering error, the proposed event-driven encrypted control framework ensures the asymptotic stability of the overall closed-loop system. Finally, a simulation example of the secure motion control for an inverted pendulum cart system is presented to evaluate the effectiveness of the theoretical results.
Encrypted control has been extensively studied to ensure the confidentiality of system states and control inputs for networked control systems. This paper presents a computationally efficient encrypted control framework for networked systems enabled by quantum communication. A quantum channel between sensors and actuators is used to generate identical secret keys, whose security is further enhanced through quantum key distribution. These keys enable lightweight encryption and decryption while preserving confidentiality and control accuracy. We develop a novel encryption-decryption architecture for state-feedback control of linear systems based on quantum keys, and characterize the impact of quantum state errors on closed-loop stability. In particular, we establish the existence of a critical threshold on intrinsic quantum noise below which stability is guaranteed. In contrast to classical encrypted control schemes, which may collapse under a single key-bit error, the proposed quantum encrypted control exhibits strong robustness to key imperfections. We further adopt quantization techniques to address the scenarios with limited communication bits in practical situations, and implement privacy protection for quantum keys based on a stochastic quantizer. These results demonstrate that integrating quantum technologies into control systems in a nontrivial and principled manner, even at their current level of maturity, can yield substantial performance gains in reducing computational complexity and improving resilience to key errors while ensuring security against multiple eavesdropping sources.
We consider a problem of implementing dynamic controllers over encrypted data for asymptotic stabilization of closed-loop systems. Though a time-varying quantizer is used and it can be infinitesimally fine with time, a major issue is that the underlying space for encrypted messages is unavoidably finite and the controller receives a limited amount of quantized data. To resolve this issue, the proposed method takes advantage of the state matrix consisting of integers, which enables the controller to generate only lower bits of the same output without computing the upper bits. Whenever a portion of the upper bits of output has converged, the computation scope can be moved further lower, receiving only lower bits of the measurement. The quantization is scheduled and the size of the message space is predetermined from the convergence rate, so that the feedback input is restored from the outcome of the lower bits, no matter how fine quantization is performed in the end. As a consequence, asymptotic stabilization can be achieved by encrypted operation, despite the limited controller capacity.
The design of encrypted controllers, which perform control operation directly over encrypted signals via homomorphic cryptosystems, should consider both security of the control data and performance of the controller. Considering the use of Learning With Errors (LWE) based cryptosystem, in this paper, we present a design procedure for encrypted linear dynamic controllers. Providing a guideline for choosing the parameters of the cryptosystem as well as quantization, the procedure guarantees both the desired level of security and performance. Receiving the encrypted signals of the plant input as well as the output, the proposed controller is able to perform the dynamic operation over encrypted data for infinite time horizon, without the use of decryption or reset of the state. Thanks to additively and multiplicatively homomorphic property of the LWE-based cryptosystem, information of both control signals and gain matrices is protected by encryption.
This paper proposes an encrypted control algorithm to the average consensus problem for agents whose communication topology is represented by a strongly connected digraph. Unlike many applications that use encryptions, encrypted control systems require encryptions and decryptions repeatedly at every sampling time. Thus, it is beneficial if the control algorithm allows us to balance the cipher strength and processing time. This property is acquired by constructing a finite-level uniform quantizer whose sensitivity changes with the evolution of the system, designing its associated control law, and then combining it with the Paillier cryptosystem. Numerical analysis of the quantized control law is also presented.
In this paper we develop a simple, yet effective, secure communication protocol based on linear algebra that allows messages to be transmitted securely, without the need to encrypt them through computationally demanding approaches or to consider integer-valued plain messages, except during an initialization phase. The scheme is composed by several layers of protection and is applied to a scenario involving a networked Kalman filter fed by measurements sent by a sensor through a public network. Specifically, the two actors interact at initialization by exchanging suitably encrypted matrices; then, each message is split in two parts and is altered via a linear non-invertible transformation. Notably, each message is guaranteed to correspond to an unobservable measurement, while the receiver is provided with a strategy to reconstruct the message through simple calculations based on the information exchanged during the initialization phase. To further mask the messages, Gaussian noise with statistics known to both parties is artificially added to the messages. A simulation campaign completes the paper and demonstrates its effectiveness experimentally.
We introduce for non-uniform messages a novel hybrid universal network coding cryptosystem (NU-HUNCC) in the finite blocklength regime that provides Post-Quantum (PQ) security at high communication rates. Recently, hybrid cryptosystems offered PQ security by premixing the data using secure coding schemes and encrypting only a small portion of it, assuming the data is uniformly distributed. An assumption that is often challenging to enforce. Standard fixed-length lossless source coding and compression schemes guarantee a uniform output in normalized divergence. Yet, his is not sufficient to guarantee security. We consider an efficient almost uniform compression scheme in non-normalized variational distance for the proposed hybrid cryptosystem, that by utilizing uniform sub-linear shared seed, guarantees PQ security. Specifically, for the proposed PQ cryptosystem, first, we provide an end-to-end coding scheme, NU-HUNCC, for non-uniform messages. Second, we show that NU-HUNCC is information-theoretic individually secured (IS) against an eavesdropper with access to any subset of the links. Third, we introduce a modified security definition, individually semantically secure under a chosen ciphertext attack (ISS-CCA1), and show that against an all-observing eavesdropper, NU-HUNCC satisfies its conditions. Finally, we provide an analysis that shows the high communication rate of NU-HUNCC and the negligibility of the shared seed size.