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Analysis of Dynamic-Key LWE-Based Encrypted Control Systems for Asymptotic Stability and Numerical Safety
Jungjin Park, Kiminao Kogiso
TL;DR
Secure encrypted control for cyber-physical systems must address cyberattack risks while maintaining stability despite quantization and encryption errors. The paper analyzes dynamic-key LWE-based state-feedback control with time-varying encoders and decoders, deriving gain conditions that ensure asymptotic stability and numerical safety. A numerical example demonstrates both properties, while broader practical implementation and more general settings remain future work.
Problem
Encrypted control must protect cyber-physical-system communications while addressing stability challenges caused by quantization and encryption errors.
Method
The paper uses a Lyapunov-based analysis of dynamic-key LWE encrypted state-feedback control with static controller-gain and time-varying state quantizers.
Results
A numerical example demonstrates that the proposed time-varying encoders and decoders achieve both asymptotic stability and numerical safety.
Takeaways & Limitations
The derived quantization-gain conditions provide a basis for reliable encrypted state-feedback control that avoids overflow while accounting for encryption errors.
Takeaways & Limitations
Practical implementation in real systems and extension to more general control settings remain future work.
Abstract
from arXiv · showhide
This study analyzes dynamic-key Learning-with-Errors (LWE)-based encrypted state-feedback control systems with time-varying encoders and decoders. Using a Lyapunov-based approach, we derive conditions on the time-varying encoder and decoder parameters that ensure both asymptotic stability and numerical safety by preventing overflow. The validity of the derived conditions is confirmed through numerical examples.
1. INTRODUCTION
The paper motivates secure encrypted control for cyber-physical systems, where connectivity creates cyberattack risks and stability remains challenging under quantization and encryption errors. It proposes dynamic-key LWE-based encoders and decoders with gain conditions for asymptotic stability and overflow prevention.
- Cyber-physical systems connect physical plants and networked controllers, but this connectivity increases risks to measurement and control channels.
- Encrypted control uses homomorphic encryption to protect controller parameters and transmitted signals while enabling computation on encrypted data.
- Prior work treated encryption and quantization errors as bounded perturbations, but static quantizers alone may not achieve asymptotic stability.
- The study derives quantization-gain conditions using a Lyapunov function to ensure asymptotic stability despite quantization and encryption errors.
- The admissible gain range is characterized to provide numerical safety by preventing overflow in the encrypted state-feedback control system.
2. PRELIMINARIES
The preliminaries establish notation, digit decomposition, and the dynamic-key LWE encryption scheme used for encrypted control. They describe key and ciphertext updates, homomorphic operations, and the associated encryption noise.
- 2.1. Notations: The notation defines real, rational, integer, natural-number, plaintext, and ciphertext spaces as R, Q, Z, N, M, and C, respectively.
- 2.1. Notations: The digit decomposition map G−1 expands each entry of a vector into base-ν digits, with w = G G−1(w).
- 2.2. Dynamic-Key LWE Encryption Scheme: The dynamic-key LWE encryption scheme is defined at step k as a tuple containing conventional encryption algorithms and key or ciphertext update maps.
- 2.2. Dynamic-Key LWE Encryption Scheme: The setup and encryption algorithms choose moduli, noise parameters, private keys, vector ciphertexts, and matrix ciphertexts for plaintext elements.
- 2.2. Dynamic-Key LWE Encryption Scheme: Key and ciphertext update maps operate at each step using random vectors, while homomorphic addition and multiplication combine ciphertexts.
- 2.2. Dynamic-Key LWE Encryption Scheme: Homomorphic operations increase encryption noise, with addition accumulating errors and multiplication producing an error term involving plaintext and digit-decomposed ciphertext components.
3. PROBLEM DESCRIPTION
The paper formulates reliable encrypted state-feedback control using dynamic-key LWE encryption, focusing on quantization gains that jointly ensure asymptotic stability and numerical safety. The system must account for quantization and encryption errors while preventing plaintext overflow.
- Encoder and decoder: Real-valued states and feedback gains are quantized with encoder-decoder maps before encrypted control computations.The gain uses a static gain γ0, while the state is encoded at every step with a time-varying gain γ1(k).
- Numerical safety: Plaintext values must remain within M = Zt because exceeding its range causes modulo-t overflow and incorrect decoding.
- Controller encryption: The encrypted controller maps ciphertexts of the gain and state to a ciphertext of the control input using an updatable private key.The gain ciphertext is initialized from the encoded gain, while state ciphertexts are generated at each step using the current key.
- Problem statement: Quantization-induced state error and encryption-induced control error can affect performance and lead to instability or overflow.The paper therefore analyzes both error types when deriving conditions on γ0 and γ1.
- Problem formulation: Reliable encrypted control requires both asymptotic stability under errors and no overflow during operation.
4. ERROR BOUND ANALYSIS
The error-bound analysis characterizes quantization and encryption errors before studying their effects on the encrypted control system. The encryption bound uses bounded noise tails and aggregates contributions from matrix operations and multiplications.
- Quantization error is bounded under equal encoder gains γ0 = γ1 ≡ γ̄.
- Encryption error is bounded for given parameters p, with the noise tail controlled by choosing κ so Pr(|ω| > κσ) is negligible.
- The encryption-error bound combines matrix norms, dimension-dependent terms, noise scale σ, and the number of np multiplications.The derivation bounds each noise contribution and then sums the resulting terms.
- These quantization and encryption error bounds are used to analyze their effects on the encrypted control system.
- Unlike cyclic-group plaintexts in ElGamal, LWE schemes use uniformly spaced Zt plaintext elements with unit adjacent width.
5. CONDITION FOR RELIABILITY
The section derives quantization-gain conditions that make the dynamic-key LWE encrypted control system reliable, combining asymptotic-stability requirements with bounds that prevent overflow and underflow. A Lyapunov analysis establishes the stability result, while separate gain bounds ensure numerical safety.
- 5. CONDITION FOR RELIABILITY: The static controller-gain encoder and time-varying state encoder require admissible quantization-gain ranges for reliable encrypted state-feedback control.The analysis treats γ0 for the controller gain and γ1(k) for the system state.
- 5.1. Main Result: The lower bound on γ0 ensures the encoded feedback matrix is Schur stable, while an upper bound prevents overflow of its encoded entries.Lemma 3 establishes Schur stability under condition (10a); the overflow argument bounds each encoded gain entry.
- 5.2.2. Quantization error on communication signals: Under a Schur-stable encoded feedback matrix, conditions on γ1(k) ensure asymptotic stability despite state quantization and encryption errors.The Lyapunov function decreases when the state norm exceeds a γ1(k)-dependent bound, and the resulting state norm converges to zero.
- 5.2.2. Quantization error on communication signals: As the state norm decreases toward zero, γ1(k) must increase over time to preserve the inequality supporting asymptotic stability.This is the dynamic-quantization behavior identified in Remark 3.
- 5.1. Main Result: The stability analysis accounts for encryption error δu, while Lemmas 5 and 6 provide the study’s original results for communication-signal quantization and numerical safety.The paper also states that the analysis applies to static-key LWE encryption because its encryption-error bound is independent of time.
6. NUMERICAL EXAMPLE
The numerical example compares the proposed time-varying encoders with static encoders and prior time-varying encoders, while measuring signal responses and computation time.
- 6. NUMERICAL EXAMPLE: The proposed experiment uses a static controller quantizer and a state-dependent time-varying quantization gain updated from the measured state norm.γ1(k) is updated at each step using the measured ∥x(k)∥ to satisfy Theorem 1 with bias 0.01.
- 6.2. Results: The simulation compares proposed time-varying encoders, static encoders, and the time-varying encoders from.Fig. 2 presents the proposed-versus-static comparison, while Fig. 3 reports results for.
- 6.2. Results: The plotted responses cover δu, the states x1, x2, and x3, the input u, and the quantization gain γ1.The original system is included as a reference in the state and input responses.
- 6.2. Results: The implementation measured a total processing time of (7.51 ± 0.07) × 10^-1 ms per control step.The total includes encoding, encryption, homomorphic operations, decryption, decoding, key update, and ciphertext update.
- 6.2. Results: The reported processing time suggests potential real-time implementation with millisecond-scale sampling periods.This implication is stated for the proposed dynamic-key LWE-based encrypted controller.
7. CONCLUSION
The study derives quantization-gain conditions for dynamic-key LWE-based encrypted state-feedback systems and uses a numerical example to demonstrate asymptotic stability and numerical safety. Practical implementation and extension to more general settings remain future work.
- 7. CONCLUSION: The study derives conditions on quantization gains that guarantee asymptotic stability and characterize admissible ranges for avoiding overflow.These conditions address reliability of encrypted state-feedback control with time-varying encoders and decoders.
- 7. CONCLUSION: A numerical example demonstrates that the proposed time-varying encoders and decoders achieve both asymptotic stability and numerical safety.Numerical safety is defined here through avoidance of overflow.
- 7. CONCLUSION: Future work will investigate practical implementation in real systems and extend the framework to more general control system settings.