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

2025-04-12

AI Summary

In homomorphic encryption-based dynamic control, existing approaches struggle to simultaneously guarantee asymptotic stability and computational efficiency, particularly due to the need for frequent re-encryption of control inputs. Method: This paper proposes an encrypted control design framework that eliminates re-encryption by introducing a dynamic quantization scaling factor. This factor enables integer-valued representation of continuous-domain controller coefficients while decoupling scaling factor design from closed-loop convergence rate—thereby removing the strong dependence of stability margins on quantization parameters inherent in prior methods. Contribution/Results: The framework is the first to enable direct adaptation of arbitrary pre-specified controllers under finite-modulus homomorphic encryption. It ensures asymptotic stability of the encrypted closed-loop system, prevents quantizer saturation, and establishes implementable upper bounds on the scaling factor and lower bounds on the ciphertext modulus—facilitating practical deployment.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

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.

Problem

• Stabilize systems under homomorphic encryption without re-encryption
• Design controllers decoupled from closed-loop convergence rates
• Ensure asymptotic stability and prevent quantizer saturation

Innovation

• Homomorphic encryption without re-encrypting inputs
• Dynamic quantization scaling controller coefficients
• Decoupled zooming-in factor for stability