Securing Control Systems with Polynomial-Packed Encryption

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Author: Denis Avetisyan

A new approach leverages rational canonical form to significantly accelerate encrypted control designs based on Ring-LWE, enhancing both speed and efficiency.

This work presents an efficient encrypted controller design utilizing rational canonical form and polynomial packing to reduce computational complexity for recursive multiplication in Ring-LWE based homomorphic encryption.

Securing control systems against adversarial attacks necessitates cryptographic solutions, yet fully homomorphic encryption often introduces substantial computational overhead. This paper, ‘Taking Advantage of Rational Canonical Form for Faster Ring-LWE based Encrypted Controller with Recursive Multiplication’, addresses this challenge by presenting an efficient encrypted controller design leveraging the Ring-LWE cryptosystem. Through a system-theoretical approach and the strategic application of rational canonical form alongside polynomial packing, the design significantly reduces both the time and space complexity of recursive multiplication operations. Could this method pave the way for practical, real-time implementation of privacy-preserving control in critical infrastructure?

The Inevitable Vulnerability of Control Systems

Modern networked control systems, while offering unprecedented levels of automation and efficiency, are becoming increasingly susceptible to adversarial attacks. As these systems permeate critical infrastructure – from power grids and water treatment facilities to transportation networks and manufacturing plants – the potential consequences of malicious interference escalate dramatically. These attacks aren’t limited to simple disruption; adversaries can manipulate sensor data, inject false commands, or even compromise the entire control infrastructure, leading to physical damage, economic loss, and safety hazards. Consequently, a pressing need exists for robust security measures that go beyond traditional cybersecurity protocols, demanding innovative approaches to detect, prevent, and mitigate these evolving threats and ensure the reliable operation of these vital systems.

The implementation of conventional encryption techniques in networked control systems frequently encounters a significant bottleneck: computational overhead. While designed to safeguard data integrity and confidentiality, algorithms like RSA and AES demand substantial processing power, especially when applied to the continuous stream of data inherent in real-time control loops. This burden can introduce unacceptable latency, disrupting the timely execution of control commands and potentially destabilizing the entire system. Consider a self-driving car or a smart grid; even slight delays in processing sensor data and issuing corrective actions could have catastrophic consequences. Consequently, a direct application of these established cryptographic methods often proves impractical, necessitating innovative approaches that minimize computational cost without compromising security – a challenge driving research into alternative encryption schemes and hardware acceleration techniques.

Contemporary control systems increasingly demand the safeguarding of sensitive data – from critical infrastructure telemetry to proprietary algorithms – which is driving a fundamental shift in how control tasks are executed. Historically, data was decrypted for processing, creating a vulnerability point; however, emerging techniques now prioritize computations directly on encrypted data, a field known as homomorphic encryption and secure multi-party computation. This paradigm allows control logic to operate on ciphertext without prior decryption, preserving data confidentiality throughout the entire control loop. Such an approach not only mitigates the risk of data breaches and malicious interference but also enables collaborative control scenarios where multiple parties can contribute data or algorithms without revealing their underlying information, opening avenues for more resilient and trustworthy automated systems. The advancement of these cryptographic tools promises a future where security and functionality are no longer competing priorities in the design of networked control systems.

Ring-LWE: A Necessary Foundation, But Don’t Expect Miracles

Ring-LWE (Learning With Errors over Rings) provides a robust basis for constructing homomorphic encryption schemes. These schemes allow for computations to be performed directly on encrypted data without requiring decryption first. The security of Ring-LWE relies on the presumed hardness of distinguishing between uniformly random polynomials and those generated by adding a small error term, defined over a specific polynomial ring. This allows ciphertext to be mathematically manipulated – added, multiplied, etc. – and upon decryption, the result corresponds to the operation performed on the original plaintext. The use of a ring structure, rather than traditional lattices, significantly improves computational efficiency, particularly for large-scale encrypted computations. Specifically, the ring structure enables the use of Fast Fourier Transforms (FFTs) for efficient polynomial multiplication, a core operation in many homomorphic encryption schemes.

Ring-LWE computations are performed within a polynomial ring, denoted as R = \mathbb{Z}_q[x]/(f(x)), where q is a prime modulus and f(x) is an irreducible polynomial. This ring structure allows for efficient arithmetic operations due to the reduction modulo f(x), limiting the degree of polynomials and preventing unbounded growth of coefficients. Specifically, polynomial multiplication in this ring can be performed in O(n log n) time using the Number Theoretic Transform (NTT), where n is the degree of the polynomial, which is significantly faster than the O(n^2) complexity of standard polynomial multiplication. The algebraic properties of the ring enable the use of these optimized algorithms for both encryption and decryption processes, contributing to the overall performance of the Ring-LWE scheme.

Optimized implementation of Ring-LWE relies on techniques that reduce computational complexity within the encrypted domain. The ‘External Product’ is a specialized multiplication operation defined for polynomial rings, enabling efficient computation of a \cdot b where a and b are polynomials, significantly faster than standard polynomial multiplication. Furthermore, ‘Automorphism’ involves applying a ring isomorphism – a mapping that preserves the ring structure – to ciphertexts. This allows for the introduction of variability and can be used to mask data, mitigate certain attacks, and reduce the growth of noise during homomorphic computations, thereby extending the number of operations that can be performed before decryption errors occur.

Algorithms Bend, But They Don’t Break: Optimizing Control for Encryption

Dynamic control systems frequently utilize recursive multiplication operations to update state variables over time. These operations, while efficient in traditional computing environments, present a significant challenge when implemented within a Homomorphic Encryption (HE) scheme. HE introduces substantial computational overhead due to the encrypted nature of the data; each multiplication requires operations on ciphertexts rather than plaintexts. This dramatically increases the computational cost, particularly as the number of recursive iterations grows. The complexity stems from the need to maintain the encrypted state throughout the entire control loop, with each multiplication contributing to a cumulative performance bottleneck. Consequently, minimizing the number of recursive multiplications, or finding alternative formulations, is crucial for practical HE-based control system implementations.

Transforming control algorithms into Rational Canonical Form involves a change-of-basis operation on the system’s State Matrix, A. This transformation yields a block-diagonal matrix composed of companion matrices, each corresponding to an eigenvalue of A. By decoupling the system’s dynamics into these simpler, independent blocks, the computational complexity of recursive operations – particularly matrix multiplications inherent in control law evaluation – is significantly reduced. The resulting simplified State Matrix requires fewer multiplications and additions when performing calculations within the Homomorphic Encryption domain, leading to improved performance compared to operating directly on the original, potentially full-rank, State Matrix. This reduction in complexity is achieved without altering the underlying system dynamics, ensuring the control algorithm’s functionality remains consistent.

A Circulant Matrix is a specific type of matrix where each row is a circular shift of the first row, offering significant computational advantages within homomorphic encryption (HE) schemes. This structure allows for the efficient implementation of polynomial multiplications using the Fast Fourier Transform (FFT), reducing the complexity of operations like matrix-vector multiplication from O(n2) to O(n log n). Within the HE domain, this translates to fewer ciphertext multiplications and a reduction in noise growth. The algebraic properties of circulant matrices enable the decomposition of complex matrix operations into a series of simpler, more manageable calculations, directly impacting the performance of control algorithms implemented with HE.

Maintaining an integer state matrix is paramount when implementing control algorithms with homomorphic encryption (HE) due to the computational cost of bootstrapping. Bootstrapping, necessary when HE ciphertext grows beyond manageable limits, is significantly more expensive than basic HE operations. Prior approaches to HE control suffered from a computational complexity of O(n) related to the size of the state matrix. Our method, by ensuring all state matrix elements remain integers, avoids repeated bootstrapping and achieves a reduced computational complexity of O(log\ n), representing a substantial improvement in efficiency for large state spaces.

The Devil is in the Details: Efficient Encoding and System Security

The efficiency of modern cryptographic systems often hinges on minimizing computational load, and the ‘Packing Method’ offers a significant advancement in this regard. Instead of processing each data point individually, this technique encodes multiple values into a single polynomial representation. This consolidation allows for parallel computation within the Ring-LWE scheme, effectively amortizing the cost across numerous data points. Consequently, operations that would typically require n separate calculations can be performed on a single polynomial, drastically reducing the overall computational burden and enabling faster processing speeds. The method cleverly leverages the algebraic structure of polynomial rings to achieve this compression, proving particularly beneficial in resource-constrained environments or applications demanding real-time performance.

The efficiency of computations within the Ring-LWE cryptosystem heavily relies on maintaining integer values throughout the encoding process, and a carefully selected ‘Scaling Factor’ is paramount to achieving this. Without this factor, intermediate calculations can introduce floating-point numbers, drastically increasing computational complexity and negating the benefits of the Ring-LWE approach. This scaling ensures that all data, even after undergoing polynomial packing and other transformations, remains within the integer domain, allowing for the use of fast and efficient integer arithmetic. Specifically, the scaling factor determines the magnitude of the encoded data, preventing potential overflows or underflows during calculations and guaranteeing the correctness of the cryptographic operations. By proactively controlling the data range, the scaling factor directly contributes to both the speed and security of the entire system, enabling practical implementations of encrypted controllers.

The robustness of data protection within the Ring-LWE cryptosystem is fundamentally dictated by the chosen ‘Security Parameter’. This parameter-typically denoted as λ-effectively establishes the computational difficulty for potential adversaries attempting to decrypt the encoded information. A larger security parameter necessitates significantly more computational resources for any attack, thereby enhancing security, but also increasing the overhead associated with both encryption and decryption processes. Conversely, a smaller parameter offers improved performance and reduced computational cost, albeit at the expense of diminished security. This inherent relationship provides a crucial tunable trade-off, allowing system designers to precisely balance the desired level of protection against the available computational resources and real-time performance requirements of the application. Careful selection of this parameter is therefore paramount in establishing a practical and effective security posture.

The culmination of efficient data encoding and cryptographic techniques yields practical Encrypted Controllers, systems designed to operate securely without sacrificing real-time responsiveness. These controllers achieve a remarkably low computational and memory complexity of O(log\ n), meaning processing demands scale favorably with input size n. This efficiency is further underscored by a minimized number of external products-the core operation in many cryptographic schemes-calculated as 2 + \kappa(1 + log_2\ n) + \lceil log_2\ m \rceil, where κ represents the security parameter and m denotes a system-specific value. This optimized architecture allows for secure control systems that are not only theoretically sound but also realistically implementable, opening possibilities for privacy-preserving automation and secure data processing in resource-constrained environments.

The pursuit of optimized encrypted controllers, as detailed in this work leveraging rational canonical form, feels predictably Sisyphean. This paper attempts to streamline recursive multiplication within the Ring-LWE framework, promising efficiency gains. It’s a clever approach, certainly, but one built on a foundation of mathematical elegance that production systems will inevitably erode. As Jean-Jacques Rousseau observed, “The body politic, in order to be well constituted, must be proportionate to the strength of its citizens.” This holds true for cryptographic schemes as well; theoretical efficiency means little when the realities of deployment introduce overhead and unforeseen complications. The elegance of polynomial packing will, without question, become tomorrow’s tech debt.

What’s Next?

The pursuit of efficient homomorphic encryption inevitably leads to increasingly elaborate mathematical scaffolding. This work, leveraging rational canonical form to tame the complexities of recursive multiplication within Ring-LWE, is no exception. It successfully postpones, rather than solves, the inherent problem: each layer of optimization introduces new opportunities for subtle, production-level failures. The elegance of polynomial packing and canonical forms will, predictably, collide with the messy realities of implementation – imperfect noise distributions, side-channel vulnerabilities, and the sheer cost of verifying correctness at scale.

Future effort will likely focus on automating the derivation of these canonical forms, perhaps through symbolic computation systems. However, any system promising to ‘simplify’ the process merely adds another layer of abstraction – and another point of potential divergence between theory and practice. The true challenge isn’t minimizing theoretical complexity, but embracing the inevitable entropy of real-world deployments. Consider this a temporary reprieve, not a victory.

Ultimately, the field will be forced to confront the fact that ‘faster’ encryption isn’t always ‘better’ encryption. The cost of maintaining these intricate structures – the documentation (a myth invented by managers), the specialized tooling, the constant vigilance against newly discovered attacks – will eventually outweigh the performance gains. The next breakthrough won’t be a clever algorithm, but a pragmatic acceptance of limitations.

Original article: https://arxiv.org/pdf/2512.24658.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2026-01-02 08:28