Encrypted Control: From Pioneering Realizations to Modern Implementations

Concetti Chiave

This tutorial paper provides exemplary realizations of encrypted state feedback and PI controllers based on popular homomorphic cryptosystems, facilitating access to encrypted control for researchers lacking a cryptographic background.

Sintesi

This tutorial paper aims to facilitate the access to encrypted control by providing exemplary realizations based on popular homomorphic cryptosystems, such as Paillier, GSW, and CKKS.
The paper first provides a brief overview of homomorphic encryption (HE) and the key characteristics of the three cryptosystems used in the implementations. It then discusses the necessary controller reformulations required for an encrypted realization of linear state feedback and PI control.
The main part of the paper presents the actual implementations of the two control schemes using a custom Matlab toolbox developed by the authors. The state feedback controller is implemented in both partially and fully encrypted versions, while the PI controller is realized in a fully encrypted manner.
The numerical experiments demonstrate the closed-loop performance of the encrypted controllers for varying accuracy levels, showing that the effect of the scaling factor is minor for the studied control system. The paper also notes that while the multiplicative depth is finite for both schemes, the PI controller's recursive nature leads to an unbounded number of encrypted additions, which is nonetheless handled well due to the stabilized closed-loop.
The tutorial concludes with a summary and an outlook on potential extensions, such as considering more functionalities, performance optimizations, latency, or the integration of an actual cloud.

Statistiche

The state space model of the benchmark system is given by:
x(k + 1) = [-0.27 -0.24 -0.08; -0.20 -0.35 -0.17; -0.22 -0.02 -0.36]x(k) + [-0.05; -0.11; -0.41]u(k)
y(k) = [0 0 1.56]x(k)
The stabilizing state feedback gain is K = [-0.07 0.06 -0.12].
The stabilizing PI controller gains are Kp = -0.5 and Ki = -0.75.

Citazioni

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Approfondimenti chiave tratti da

A code-driven tutorial on encrypted control

by Nils... alle arxiv.org 04-09-2024

https://arxiv.org/pdf/2404.04727.pdf

Domande più approfondite

How could the encrypted control implementations be extended to handle time-varying or nonlinear systems

To extend the encrypted control implementations to handle time-varying or nonlinear systems, several approaches can be considered. One option is to incorporate adaptive control techniques that can adjust the controller parameters in real-time based on the system's varying dynamics. This adaptation can be done using encrypted computations to ensure data privacy. Additionally, techniques like model predictive control (MPC) can be utilized to handle nonlinearities by formulating the control problem as an optimization task that can be solved using encrypted data. By leveraging the capabilities of homomorphic encryption, the control algorithms can be designed to accommodate time-varying and nonlinear system behaviors while maintaining data security.

What are the potential challenges and limitations of applying fully homomorphic encryption (FHE) schemes in real-time control applications

Fully homomorphic encryption (FHE) schemes offer the capability to perform arbitrary computations on encrypted data without the need for decryption, making them highly attractive for secure control applications. However, there are several challenges and limitations when applying FHE in real-time control scenarios. One major limitation is the computational overhead associated with FHE operations, which can significantly impact the control system's response time. The complexity of FHE algorithms can lead to increased latency, making them unsuitable for time-critical control applications. Moreover, the multiplicative depth and noise growth in FHE schemes can limit the types of computations that can be efficiently performed, posing challenges for complex control algorithms. Ensuring the scalability and efficiency of FHE implementations in real-time control applications remains a key challenge that needs to be addressed.

What are the implications of using encrypted control in safety-critical systems, and how could the security-performance trade-offs be analyzed

Using encrypted control in safety-critical systems introduces both benefits and challenges. On the positive side, encrypted control can enhance data privacy and security, protecting sensitive information from unauthorized access and cyber threats. This is particularly crucial in safety-critical systems where the integrity of control data is paramount. However, there are implications related to the performance-security trade-offs inherent in encrypted control. The computational overhead of encryption and decryption processes can introduce delays in control actions, affecting the system's real-time responsiveness. Analyzing the security-performance trade-offs involves evaluating factors such as encryption/decryption speed, communication latency, and computational resources required for encrypted control. Balancing the security needs with the system's performance requirements is essential to ensure the safe and efficient operation of safety-critical systems.