Input-Output History Feedback Controller for Encrypted Control with Leveled Fully Homomorphic Encryption

TL;DR

This work tackles secure outsourcing of dynamic controllers by eliminating the need to decrypt controller states, which traditionally causes overflow and instability in encrypted control. It introduces an input/output history feedback controller (IOHFC) representation that expresses linear time-invariant controllers solely through history data, enabling stable encrypted operation with leveled BFV via CRT batching. The authors derive a stability condition under quantization and provide a bound on worst-case output degradation, validating the approach with a numerical simulation on a decentralized PI controller for a tank process. The method achieves practical real-time performance thanks to an efficient matrix-vector multiplication scheme and input re-encryption, offering a viable path for secure cloud-based control without bootstrapping. Extensions to nonlinear controllers via Koopman operator theory are proposed for future exploration.

Abstract

Protecting the parameters, states, and input/output signals of a dynamic controller is essential for securely outsourcing its computation to an untrusted third party. Although a fully homomorphic encryption scheme allows the evaluation of controller operations with encrypted data, an encrypted dynamic controller with the encryption scheme destabilizes a closed-loop system or degrades the control performance due to overflow. This paper presents a novel controller representation based on input-output history data to implement an encrypted dynamic controller that operates without destabilization and performance degradation. Implementation of this encrypted dynamic controller representation can be optimized via batching techniques to reduce the time and space complexities. Furthermore, this study analyzes the stability and performance degradation of a closed-loop system caused by the effects of controller encryption. A numerical simulation demonstrates the feasibility of the proposed encrypted control scheme, which inherits the control performance of the original controller at a sufficient level.

Figures (9)

• Figure 1: Cloud-based encrypted control system under adversaries. The blue arrows are encrypted channels, and the red arrows illustrate eavesdropping attacks.
• Figure 2: Schematic picture of the IOHFC ($L=7$).
• Figure 3: Illustration of secure matrix-vector multiplication over the ciphertext space. (a) Matrix $M\in\mathbb{Z}_{T}^{2\times3}$, vector $v\in\mathbb{Z}_{T}^{3}$, and target vector $[Mv]_{T}\in\mathbb{Z}_{T}^{2}$. (b) The elements of $M$ and $v$ are embedded in corresponding $N$-dimensional vectors $z_{1}$ and $z_{2}$, respectively. The white boxes of the vectors contain zero. Element-wise multiplication $z_{1}\odot z_{2}$ between the vectors is computed. (c) The computed vector is added with the rotation of itself three times to obtain $z_{3}$. The gray boxes of $z_{3}$ are wasted data. The target vector is constructed from the first and forth elements of $z_{3}$.
• Figure 4: Schematic picture of the modified IOHFC ($L=7$).
• Figure 5: Encrypted control system with IOHFC.

Theorems & Definitions (15)

• Theorem 1
• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Lemma 1
• proof
• Theorem 2
• proof
• Theorem 3