⚡ electrical engineering

Dynamic-Key Post-Quantum Encrypted Control Against System Identification Attacks

This paper proposes a post-quantum encrypted control method using a dynamic-key Learning with Errors (LWE) scheme that updates both keys and ciphertexts to prevent system identification attacks while ensuring decryption accuracy through analyzed error growth and parameter conditions.

Original authors: Jungjin Park, Kiminao Kogiso

Published 2026-04-28

📖 3 min read☕ Coffee break read

CC BY 4.0

Original authors: Jungjin Park, Kiminao Kogiso

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are the manager of a high-tech, automated smart city. This city is run by a "brain" (a controller) that sends secret instructions to various machines (the system) to keep everything running smoothly—like adjusting power grids or traffic lights.

The problem is that hackers are getting smarter. They aren't just trying to break in; they are trying to "spy" on the instructions to learn exactly how your city works. If they figure out the "rhythm" of your machines, they can launch a "stealth attack"—a way to sabotage the city that looks perfectly normal to your sensors but actually causes a massive crash later.

This paper proposes a new way to protect the city using "Quantum-Proof Shape-Shifting Codes."

Here is the breakdown of how it works:

1. The Problem: The "Pattern" Trap

Traditional encryption is like a high-quality padlock. It’s very strong, but if a hacker watches you use the same padlock every day, they can eventually study the shape of the key and the sound of the click. In technical terms, if the encryption stays the same (a "static key"), a hacker can collect enough data to perform "System Identification"—basically, they build a digital twin of your city to predict your every move.

2. The Solution: The Shape-Shifting Key (Dynamic-Key LWE)

The researchers created a system based on something called LWE (Learning with Errors). Think of LWE as a math puzzle that is intentionally "fuzzy." Instead of giving a perfect answer, the system gives an answer that is almost right, but has a tiny bit of intentional "noise" or "static" added to it. Even a super-powerful quantum computer finds it nearly impossible to solve this fuzzy puzzle.

But they took it a step further with Dynamic Keys.

The Analogy: Imagine you are sending secret messages to a friend using a codebook.

Old Way: You both use the same codebook every day. Eventually, a spy learns the code.
The Paper's Way: Every single time you send a message, the codebook itself changes. You don't just change the message; you change the entire language you are speaking, and you do it so fast that by the time the spy learns "Hello," you are already speaking a completely different dialect.

3. The Challenge: The "Static" Problem

There is a catch. Because this system uses "fuzzy" math (adding noise to stay secure), that noise tends to grow every time you do a calculation.

The Analogy: Imagine you are trying to play a game of "Telephone." You whisper a message to a friend, but there is a little bit of static on the line. Then they whisper it to the next person, and more static is added. If you do this too many times, the message becomes complete gibberish.

The researchers did the heavy math to figure out exactly how much "static" they can allow before the message breaks. They created a "Design Manual" (a set of mathematical rules) that tells engineers exactly how to pick their settings so that the message stays perfectly clear for the machines, even though it's heavily encrypted and constantly changing.

Summary: Why does this matter?

In short, this paper provides a blueprint for building industrial systems (like power plants or smart cities) that are:

Quantum-Proof: Safe even from the terrifyingly fast computers of the future.
Spy-Proof: Because the "key" changes every second, hackers can't collect enough data to learn how the system works.
Reliable: It ensures that despite all the complex math and "noise," the machines still receive the exact, correct instructions they need to operate safely.

Technical Summary: Dynamic-Key Post-Quantum Encrypted Control Against System Identification Attacks

1. Problem Statement

As Cyber-Physical Systems (CPSs) become integral to critical infrastructure (Industry 5.0), they face increasing threats from cyber-attacks. While homomorphic encryption (HE) has been used to protect control signals, two major vulnerabilities remain:

Quantum Vulnerability: Conventional public-key cryptosystems (RSA, ElGamal) are susceptible to quantum computing attacks.
System Identification Attacks: Even with Post-Quantum Cryptography (PQC), an adversary can collect encrypted input-output data over time. If the encryption key is static, the adversary can eventually decrypt enough samples to estimate the system's internal dynamics (e.g., via least-squares), enabling sophisticated "stealthy" attacks like zero-dynamics attacks.

The core challenge is to design a control system that is both quantum-resistant and resistant to data-driven system identification by preventing the accumulation of usable encrypted data.

2. Methodology

The authors propose a novel dynamic-key Learning with Errors (LWE)-based encryption scheme integrated into a state-feedback control loop.

A. Dynamic-Key LWE Scheme:
Unlike static schemes, the proposed framework introduces update transition maps ( $T_K, T_{C_v}, T_{C_G}$ ) that simultaneously update the private key and the ciphertexts at every control step $k$ .

Key Update ( $T_K$ ): The private key $s_k$ is updated by adding a random vector $s(k)$ , ensuring that the key used at step $k$ is different from step $k+1$ .
Ciphertext Update ( $T_{C_v}, T_{C_G}$ ): The ciphertexts are updated in a way that remains consistent with the new private key, allowing the controller to perform homomorphic operations without needing to decrypt and re-encrypt at every step.

B. Encrypted Controller Design:
The controller performs computations (homomorphic addition and multiplication) directly on the encrypted state $c_x(k)$ and the encrypted gain $C_F(k)$ . The control law is implemented as:
$u(k) = \sum_{j=1}^{n_p} C_{F_j}(k) \otimes c_{x_j}(k)$
This allows the control signal to be generated in an encrypted domain.

C. Error Analysis and Parameter Design:
A critical technical hurdle in HE is the growth of noise (errors) during homomorphic operations. The authors derive a sufficient condition (Lemma 4) to ensure that the total error $e_{total}$ does not exceed the decryption threshold. They establish that if the parameters $\Delta, t,$ and $q$ satisfy a specific inequality involving the system gain and the LWE noise, the encrypted system becomes mathematically equivalent to a quantized version of the original unencrypted system.

3. Key Contributions

Extension of Dynamic-Key Control to PQC: The authors successfully transition dynamic-key encryption from the classical ElGamal scheme to the post-quantum LWE framework.
Simultaneous Update Mechanism: Development of transition maps that jointly update keys and ciphertexts, preventing adversaries from accumulating meaningful data for system identification.
Security-Driven Design Procedure: A formal methodology to select security parameters based on sample-identifying complexity (how many samples an attacker needs) and quantum sample-deciphering time (how long it takes a quantum computer to break those samples).
Error Bound Derivation: Mathematical proof that the error growth in the dynamic-key LWE controller can be suppressed to ensure correct decryption.

4. Results

The paper validates the theory through a numerical simulation of a discrete-time linear plant:

Security Verification: By setting a target operational lifetime ( $T_c = 10$ years) and a required accuracy, the authors designed a system with a quantum bit-security of $\lambda_Q = 74$ bits. The calculated quantum deciphering time exceeded the system's lifetime ( $T_Q > T_c$ ), proving the system is secure against the modeled attack.
Control Accuracy: The simulation compared the encrypted controller's output to a quantized controller. When the proposed parameter conditions were met, the outputs coincided perfectly. When the conditions were violated, significant errors were observed, confirming the necessity of their design procedure.

5. Significance

This work provides a robust blueprint for securing critical CPS infrastructure against the next generation of threats. By combining Post-Quantum Cryptography with Dynamic-Key updates, the authors move beyond simple signal protection toward system-level security, ensuring that even if an adversary intercepts data, they cannot gain enough information to model the system or launch destabilizing attacks. This is a vital step toward the realization of secure, long-term autonomous industrial systems.