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Encrypted Control Using Modified Learning with Errors-based Schemes

Robert s Lois, Daniel G. Cole

Published Jan 1, 2024• 0 citations
Publisher

Abstract

Cyber-physical systems (CPSs) require reliable, safe, and secure control of critical infrastructure, combining computational and networking capabilities, which heighten the risk of cyberattacks. These attacks can disrupt the physical process, causing unforeseen consequences. One solution is using fully homomorphic encryption (FHE) to protect the control loop, allowing for secure computations and communications without compromising signal and control system privacy. One challenge with FHE, however, is its requirement for inputs to be integers. This paper introduces a modified FHE scheme based on the learning with errors (LWE) problem. Our proposed scheme leverages a generalized LWE encoding function and modifies the Gentry-Sahai-Waters (GSW) gadget decomposition tool to encrypt the control system. Using the modified LWE scheme, we formalize a fully encrypted control system, supported by simulated results.

DOI
10.1016/j.ifacol.2024.12.013

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Cyber-physical systems (CPSs) require reliable, safe, and secure control of critical infrastructure, combining computational and networking capabilities, which heighten the risk of cyberattacks. These attacks can disrupt the physical process, causing unforeseen consequences. One solution is using fully homomorphic encryption (FHE) to protect the control loop, allowing for secure computations and communications without compromising signal and control system privacy. One challenge with FHE, however, is its requirement for inputs to be integers. This paper introduces a modified FHE scheme based on the learning with errors (LWE) problem. Our proposed scheme leverages a generalized LWE encoding function and modifies the Gentry-Sahai-Waters (GSW) gadget decomposition tool to encrypt the control system. Using the modified LWE scheme, we formalize a fully encrypted control system, supported by simulated results.
Cyber-physical systems require reliable, safe, and secure control of critical infrastructure, combining computational and networking capabilities, which heighten the risk of cyber attacks. These attacks can disrupt the physical process, causing unforeseen consequences. One solution is the use of fully homomorphic encryption (FHE) to protect the control loop, allowing for secure computations and communications without compromising signal and control system privacy. The challenge with FHE, however, is its requirement for inputs to be integers. This presentation introduces a modified Learning With Errors (LWE) FHE approach that encodes control system dynamics and signals into integers. Our proposed scheme leverages a generalized LWE encoding function and modifies the Gentry-Sahai-Waters gadget decomposition tool to encrypt the control system. Using the modified LWE scheme, we formalize a fully encrypted control system, supported by simulated results.
Fully homomorphic encryption (FHE) is a form of public-key encryption that allows the computation of arbitrary functions on encrypted data without decrypting the data. As a result, it is a useful tool with numerous applications. Certificateless encryption (CLE) is a type of public-key encryption that combines the advantages of PKI-based public-key encryption with those of identity-based encryption (IBE). Thus, certificateless fully homomorphic encryption (CLFHE) has aroused considerable research interest. Recently, Chen, Hu, and Lian proposed a leveled certificateless homomorphic encryption (CLHE) scheme and proved its semantic security based on the learning with errors (LWE) problem in the random oracle model. However, their scheme supports only homomorphic addition, but not homomorphic multiplication. In this work, we construct two leveled CLFHE schemes using the approximate eigenvector method presented by Gentry, Sahai, and Waters. Based on the hardness of the LWE problem, we prove that one scheme satisfies adaptive semantic security and anonymity in the random oracle model, whereas the other satisfies selective semantic security and anonymity in the standard model.
Fully homomorphic encryption (FHE) has important applications in cloud computing. However, almost all fully homomorphic encryption schemes share two common flaws that they all use large-scale secret keys and some operations are inefficient. In this paper, the 'special b' variant of the learning with errors problem (bLWE) is presented, and helps us construct the first circularly secure key switching process which can replace the key switching process and similar re-linearisation process used by the existing FHE schemes. Then, we present an efficient FHE. Compared with Brakerski's scheme, our scheme reduces L secret keys to one and is more efficient. Finally, we prove the chosen-plaintext attack (CPA) security of the fully homomorphic scheme and the circular security of key switching process in standard model under the learning with errors problem (LWE) assumption.
Fully homomorphic encryption (FHE) has important applications in cloud computing. However, almost all fully homomorphic encryption schemes share two common flaws that they all use large-scale secret keys and some operations are inefficient. In this paper, the 'special b' variant of the learning with errors problem (bLWE) is presented, and helps us construct the first circularly secure key switching process which can replace the key switching process and similar re-linearisation process used by the existing FHE schemes. Then, we present an efficient FHE. Compared with Brakerski's scheme, our scheme reduces L secret keys to one and is more efficient. Finally, we prove the chosen-plaintext attack (CPA) security of the fully homomorphic scheme and the circular security of key switching process in standard model under the learning with errors problem (LWE) assumption.
Fully homomorphic encryption (FHE) scheme allows for performing computations on encrypted data without decrypting it which makes it more popular in various domains. Most previous FHE constructions are limited by the large ciphertext expansion rate and error growth rate. Learning with errors (LWE) problem is a popular primitive for constructing lattice-based FHE. Many algebraic variants of LWE are also developed with further promising cryptographic features. Most previous known LWE variants (over the commutative base rings) are unified into a general framework by Peikert et al. at TCC 2019. In 2022, a new algebraically structured LWE problem over the d-degree cyclic algebra (CLWE) (with modulus q), a typical non-commutative ring, was proposed by Grover et al.. To further explore the utility and potential advantages of this new variant, it is interesting to design new lattice-based FHE schemes by using CLWE. In this paper, by following the diagrams of Gentry-Sahai-Waters (GSW13 for short), we propose a FHE scheme based on the CLWE with an even small ciphertext size. Due to our further expansion of the plaintext space, the average ciphertext expansion rate (CER for short) is smaller. More precisely, compared to original GSW13 and the related GSW-style constructions, the ciphertext size of our proposal is reduced by almost at least 23.69%, and the average CER is reduced about 96% due to our expansion of the plaintext space. Besides, our FHE has an even small error growth rate (EGR for short). After one time multiplication of two ciphertexts, the asymptotical EGR is reduced about 49.8% compared to the original GSW13 and the related GSW-style constructions. Typically, the smaller ciphertext size, CER and EGR make our FHE scheme more efficient in trusted computing environment.
Cloud computing raises new challenges for how to protect user privacy. Fully homomorphic encryption is one way to solve the problem. In this paper, we show a useful property of Dual-LWE assumption to construct Key-Switching procedure compared with LWE assumption without extra error term. Hence, we propose to construct "Errorless Key Switching" fully homomorphic encryption (FHE) scheme based on dual learning with errors (Dual-LWE) assumption. Specifically, we compile the Dual-LWE problem proposed by Gentry et.al. at STOC2008 and First-is-errorless LWE (Ferr.LWE) problem proposed by Brakerski et al. at STOC2013 into Dual-First-is-errorless LWE (Dual-Ferr.LWE) problem. Then, utilizing Dual-Ferr.LWE assumption to construct a various GPV(vGPV) scheme, and we use vGPV scheme as the fundamental building block to construct FHE with errorless key switching scheme. Lastly, under the assumption of decisional learning with errors(DLWE), we prove that our scheme is CPA secure.