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Optimal Controller and Security Parameter for Encrypted Control Systems Under Least Squares Identification

Kaoru Teranishi, Kiminao Kogiso

TL;DR

The paper addresses securing encrypted control systems against least-squares identification attacks by deriving a novel sample-identified complexity that links security to the controllability Gramian of the closed-loop system. It shows that the attacker's estimation difficulty is maximized by an optimal H2 controller that minimizes the trace of the controllability Gramian, yielding F^* = Q^* (P^*)^{-1} and the associated Gramian Psi^*. The minimum sample size N^* needed to breach a target accuracy is N^* = ⌊ n [\gamma_c tr(Psi^*)]^{-1} ⌋ + 2, and the corresponding minimum security parameter is lambda^* = ⌊ log_2 (Upsilon tau_c (N^*)^{-1}) ⌋ + 1, tying defense thresholds to attacker performance and hardware speed. The methodology enables a systematic, parameter-aware design of encrypted control systems with updatable homomorphic encryption, balancing security and computation cost, and is extensible to other attack models and encryption schemes.

Abstract

Encrypted control is a framework for the secure outsourcing of controller computation using homomorphic encryption that allows to perform arithmetic operations on encrypted data without decryption. In a previous study, the security level of encrypted control systems was quantified based on the difficulty and computation time of system identification. This study investigates an optimal design of encrypted control systems when facing an attack attempting to estimate a system parameter by the least squares method from the perspective of the security level. This study proposes an optimal $H_2$ controller that maximizes the difficulty of estimation and an equation to determine the minimum security parameter that guarantee the security of an encrypted control system as a solution to the design problem. The proposed controller and security parameter are beneficial for reducing the computation costs of an encrypted control system, while achieving the desired security level. Furthermore, the proposed design method enables the systematic design of encrypted control systems.

Optimal Controller and Security Parameter for Encrypted Control Systems Under Least Squares Identification

TL;DR

The paper addresses securing encrypted control systems against least-squares identification attacks by deriving a novel sample-identified complexity that links security to the controllability Gramian of the closed-loop system. It shows that the attacker's estimation difficulty is maximized by an optimal H2 controller that minimizes the trace of the controllability Gramian, yielding F^* = Q^* (P^*)^{-1} and the associated Gramian Psi^*. The minimum sample size N^* needed to breach a target accuracy is N^* = ⌊ n [\gamma_c tr(Psi^*)]^{-1} ⌋ + 2, and the corresponding minimum security parameter is lambda^* = ⌊ log_2 (Upsilon tau_c (N^*)^{-1}) ⌋ + 1, tying defense thresholds to attacker performance and hardware speed. The methodology enables a systematic, parameter-aware design of encrypted control systems with updatable homomorphic encryption, balancing security and computation cost, and is extensible to other attack models and encryption schemes.

Abstract

Encrypted control is a framework for the secure outsourcing of controller computation using homomorphic encryption that allows to perform arithmetic operations on encrypted data without decryption. In a previous study, the security level of encrypted control systems was quantified based on the difficulty and computation time of system identification. This study investigates an optimal design of encrypted control systems when facing an attack attempting to estimate a system parameter by the least squares method from the perspective of the security level. This study proposes an optimal controller that maximizes the difficulty of estimation and an equation to determine the minimum security parameter that guarantee the security of an encrypted control system as a solution to the design problem. The proposed controller and security parameter are beneficial for reducing the computation costs of an encrypted control system, while achieving the desired security level. Furthermore, the proposed design method enables the systematic design of encrypted control systems.
Paper Structure (13 sections, 4 theorems, 15 equations, 1 figure)

This paper contains 13 sections, 4 theorems, 15 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Homomorphic encryption
  5. Encrypted control
  6. Attack Scenario and Security Definition
  7. Attack scenario
  8. Security of encrypted control system
  9. Encrypted Control System Design
  10. Optimal controller
  11. Optimal security parameter
  12. Numerical Example
  13. Conclusions

Key Result

Lemma 1

The function is a sample identifying complexity of eq:system under the attack in Definition def:attack, where $\Psi = \Psi(F)$ is a solution to the discrete Lyapunov equation $A \Psi A^\top - \Psi + I = 0$.

Figures (1)

  • Figure 1: Estimation error and sample identifying complexity.

Theorems & Definitions (22)

  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 1
  • Definition 4
  • Remark 2
  • Definition 5
  • Remark 3
  • Remark 4
  • Definition 6
  • ...and 12 more