A dynamic-key encrypted controller remained asymptotically stable and avoided numerical overflow in a reported computer simulation, according to a preprint. The design changes the gain used to encode and decode the system’s state over time, while accounting for both quantization errors and noise introduced by encryption.
The result is conditional: Theorem 1 says the encrypted system is reliable when its specified gain conditions are met. The reported evidence consists of a mathematical analysis and one numerical example, so it does not by itself establish how the design would perform in a physical control system.
A controller designed around two kinds of error
The study asks how encoder and decoder gains can be selected for reliable encrypted state-feedback control using a dynamic-key LWE encryption scheme. Its method treats state quantization errors and encryption-induced control-input errors as separate sources of distortion that can affect stability and overflow.
The model assumes that the plant’s state can be measured, that the plant is controllable, and that the feedback gain produces a nominal closed loop that is Schur-stable, a mathematical stability condition used by the analysis. Reliability is defined in two parts: errors must not destroy asymptotic stability, and the numerical calculations must remain safe, with no overflow during operation.
To examine those conditions, the authors use a Lyapunov function, a mathematical way to track whether system behavior is moving toward stability. The error model separates state error caused by quantization from control-input error caused by encryption. The encryption-error bound also uses a constant chosen so that the probability of a large error tail is negligible.
The key is a gain that changes with the state
The central theorem gives sufficient conditions under which the encrypted system is reliable. It uses a fixed quantization gain, γ0, for the feedback gain and a time-varying gain, γ1(k), for the state. At each control step, the state gain is updated from the measured state norm with a minimum rule that includes a bias of 0.01, so the selected value continues to meet the theorem’s conditions.
For the numerical example, the reported setup used κ = 6 and encryption parameters p = (n, t, q, σ, ν, d) = (7, 240, 240, 4.0, 2, 40). It reported a security level of 128 bits and used seven ciphertexts. The fixed controller-gain quantization value in the example was γ0 = 1.1622.
The changing gains held up in one example
In the simulation, the proposed time-varying encoders and decoders were reported to achieve both asymptotic stability and numerical safety. That outcome describes one numerical example, however, and the analysis reports no formal statistical uncertainty around it.
Static encoders produced input residuals on the order of 10^-3 and state residuals on the order of 10^-4, while the fixed gain eventually fell outside the lower bound required for stability. A comparison method that chose a quantization gain without evaluating encryption noise was described as producing significant deviations from the original control signals.
The reported implementation took an average total of (7.51 ± 0.07) × 10^-1 milliseconds per control step, measured across 10,000 control steps. The report does not specify what the plus-or-minus value represents.
The result still needs a physical test
The result should be read as a theorem-based finding under stated assumptions plus a single simulation. The guarantee depends on the plant, encryption, error-bound and quantization assumptions, while the numerical result comes from one example without formal statistical uncertainty.
Physical validation, broader control settings and systematic optimization of γ0 and the γ1 update rule remain unresolved. The processing-time figure also cannot establish performance on other hardware or implementations because it was measured for the reported setup.
The document is an arXiv version 1 preprint dated 26 Aug 2026. It was supported in part by the Japan Society for the Promotion of Science through KAKENHI Grant JP26K00966.
Paper data and sources
Original title: Analysis of Dynamic-Key LWE-Based Encrypted Control Systems for Asymptotic Stability and Numerical Safety
Authors: Jungjin Park, Kiminao Kogiso
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text