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Asymptotic stabilization under homomorphic encryption: A re-encryption free method

Shuai Feng, Qian Ma, Junsoo Kim, Shengyuan Xu

TL;DR

The paper addresses secure control by encrypting a pre-given dynamic controller using additive HE without re-encrypting inputs. It introduces dynamic quantization with a zooming-in factor to scale controller coefficients to integers and shows a preliminary result requiring fast closed-loop convergence, followed by a main result that decouples convergence from the quantization scale via an observer-based architecture and additional ciphertext feedback. The contributions include explicit conditions guaranteeing a finite modulus $q$, non-saturated quantization, and asymptotic stability for the re-encryption-free encrypted controller, along with a demonstration on a batch reactor where the actuator faithfully recovers the intended control input. The work advances practical secure control by reducing reliance on re-encryption and providing rigorous bounds, enabling scalable encrypted deployment in cyber-physical systems with finite-precision cryptography.

Abstract

In this paper, we propose methods to encrypted a pre-given dynamic controller with homomorphic encryption, without re-encrypting the control inputs. We first present a preliminary result showing that the coefficients in a pre-given dynamic controller can be scaled up into integers by the zooming-in factor in dynamic quantization, without utilizing re-encryption. However, a sufficiently small zooming-in factor may not always exist because it requires that the convergence speed of the pre-given closed-loop system should be sufficiently fast. Then, as the main result, we design a new controller approximating the pre-given dynamic controller, in which the zooming-in factor is decoupled from the convergence rate of the pre-given closed-loop system. Therefore, there always exist a (sufficiently small) zooming-in factor of dynamic quantization scaling up all the controller's coefficients to integers, and a finite modulus preventing overflow in cryptosystems. The process is asymptotically stable and the quantizer is not saturated.

Asymptotic stabilization under homomorphic encryption: A re-encryption free method

TL;DR

The paper addresses secure control by encrypting a pre-given dynamic controller using additive HE without re-encrypting inputs. It introduces dynamic quantization with a zooming-in factor to scale controller coefficients to integers and shows a preliminary result requiring fast closed-loop convergence, followed by a main result that decouples convergence from the quantization scale via an observer-based architecture and additional ciphertext feedback. The contributions include explicit conditions guaranteeing a finite modulus , non-saturated quantization, and asymptotic stability for the re-encryption-free encrypted controller, along with a demonstration on a batch reactor where the actuator faithfully recovers the intended control input. The work advances practical secure control by reducing reliance on re-encryption and providing rigorous bounds, enabling scalable encrypted deployment in cyber-physical systems with finite-precision cryptography.

Abstract

In this paper, we propose methods to encrypted a pre-given dynamic controller with homomorphic encryption, without re-encrypting the control inputs. We first present a preliminary result showing that the coefficients in a pre-given dynamic controller can be scaled up into integers by the zooming-in factor in dynamic quantization, without utilizing re-encryption. However, a sufficiently small zooming-in factor may not always exist because it requires that the convergence speed of the pre-given closed-loop system should be sufficiently fast. Then, as the main result, we design a new controller approximating the pre-given dynamic controller, in which the zooming-in factor is decoupled from the convergence rate of the pre-given closed-loop system. Therefore, there always exist a (sufficiently small) zooming-in factor of dynamic quantization scaling up all the controller's coefficients to integers, and a finite modulus preventing overflow in cryptosystems. The process is asymptotically stable and the quantizer is not saturated.

Paper Structure

This paper contains 9 sections, 4 theorems, 35 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Framework
  3. Preliminaries of additively homomorphic encryption
  4. Problem formulation
  5. Controller design
  6. A preliminary result
  7. Main result
  8. Simulation
  9. Conclusions

Key Result

Lemma 1

Consider $\rho_c$ in (rho_c) and $s_F$ in (definition). If $\rho_c < s_F$, then (fourth controller) contains only integer matrices by the following steps: (a) select $\omega \in (\rho_c, s_F] \cap \mathbb Q$ such that $\frac{F}{\omega} \in \mathbb Z ^{n_x \times n_x}$, (b) select a sufficiently sma

Figures (2)

  • Figure 1: Time responses of $\log _2 \max\{\|\alpha(t)\|_\infty, \|\beta(t)\|_\infty, \|\gamma(t)\|_\infty, \\ \| \frac{C \bar{e}_o(t)}{s_1}\|_\infty \}$.
  • Figure 2: Time resporeses of $u^a(t)-u(t)$.

Theorems & Definitions (9)

  • Lemma 1
  • Proposition 1
  • Remark 1
  • Remark 2
  • Lemma 2
  • Theorem 1
  • Remark 3
  • Remark 4
  • Remark 5