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IEEE International Conference on Parallel and Distributed Systems (ICPADS)

Efficient Dynamic Multi-key FHE Scheme from LWE for Untrusted Cloud Environments

Shuchang Zeng, Ching-Fang Hsu, Jianqun Cui, Wei Chau Xie, Qihang Hou

Published Dec 17, 2023• 1 citation
Publisher

Abstract

Fully Homomorphic Encryption (FHE) provides a good solution to directly operate on the ciphertext, and the decryption result is equivalent to the corresponding operation on the plaintext. As a technique suitable for distributed environments, multi-key Fully Homomorphic Encryption (MKFHE) scheme is the most common variant of the FHE scheme since it allows encrypted data to be computed under different keys. Unfortunately, the existing dynamic MKFHE schemes based on learning with errors (LWE) still suffer from the inefficiency of long public keys, which typically grow cube in size along the lattice dimension. Moreover, there current constructions fail to provide reliable and fast algorithms to simultaneously expand ciphertexts with multiple additional keys. In order to solve the above problems, a new faster dynamic MKFHE scheme with shorter public key in asymmetric key setting from LWE is proposed in this paper, in which the size of the public key is further reduced from $\tilde O\left( {{n^3}{{(K + L)}^2}} \right)$ to $\tilde O\left( {{n^2}{{(K + L)}^2}} \right)$. In addition, our scheme cleverly adopts the dual-user cooperation method in distributed system to realize the ciphertext expansion locally, thereby reducing the computing overhead of the cloud server. More interestingly, we design a flexible parallel ciphertext expansion algorithm for the first time based on the basic algorithm. This algorithm realizes the ciphertext expansion when multiple keys are added at the same time, thus significantly improving the computational efficiency of ciphertext expansion in the dynamic MKFHE scheme. Finally, the CPA-secure of our scheme based on standard LWE assumptions is proven.

DOI
10.1109/icpads60453.2023.00153

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There has been a great deal of work on improving the efficiency of fully homomorphic encryption (FHE) scheme. Our approach, in this regard, is to use the idea of packed ciphertexts to construct a multi-bit FHE with a short public key on the basis of the learning with errors (LWE) problem. More specifically, our FHE scheme builds on a basic encryption scheme that chooses LWE samples from the Gaussian distribution and adds Gaussian error to it. This results in decreasing the number of LWE samples from 2nlogq to n + 1. We prove that our FHE scheme is pragmatically feasible and its security relies on the hardness of the LWE problem. In addition, we form a new process of key switching for multi-bit FHE based on the ideas adopted by Brakerski et al. for optimizing the process of key switching. Finally, we analyze and compare the concrete parameters between our FHE scheme and BGH13 scheme. The result shows that compared with the BGH13 scheme, our scheme has a smaller public key by a factor about logq.
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.
Abstract In dynamic multi-key Fully Homomorphic Encryption (MFHE) scheme, homomorphic operations on ciphertexts encrypted under different keys are allowed. Moreover, the resulting ciphertext can be further computed with ciphertexts under other keys. In 2016, Peikert and Shiehian presented a dynamic MFHE scheme that relies on using a Common Reference String (CRS) and circular security assumptions. Subsequently, various variants of dynamic MFHE schemes were proposed based on this scheme, but these variant schemes still rely on a CRS and circular security assumption. Therefore, constructing a MFHE GSW scheme relies solely on the LWE assumption and does not require a CRS remains a valuable challenge. This paper introduces a new dynamic MFHE scheme. Compared to existing solutions, our scheme does not require the circular security assumption and depends solely on the LWE assumption, which theoretically provides an enhanced level of security. Additionally, it eliminates the need for a shared CRS but instead replaces the CRS with each party independently generate random matrices. This decentralized approach enhances the ability of users to generate keys independently. The size of ciphertexts and public keys in our scheme is comparable to other existing solutions, thus maintaining efficiency in both computation and ciphertext extension.
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.
Abstract Traditional fully homomorphic encryption(FHE) schemes allow computation only on data encrypted under the same public key. Multi-Key Fully Homomorphic Encryption (MKFHE) enables arbitrary operations on data encrypted with different public keys, allowing all participating users jointly decrypting the final ciphertext. The multi-key BFV FHE scheme inherits BFV’s advantages in ring element encryption and scale invariance. Nonetheless, it also has some disadvantages, such as additional noise generated during the relinearization process, the need for costly transformations during the external product process, and the requirement for a Common Reference String (CRS). In this paper, we investigate the MKFHE scheme for RLWE-based BFV. Firstly, we improve the modulus size of the evaluation key and the public key to construct a modulus enchancement relinearization method, which can significantly reduce the noise generated during the relinearization process. Secondly, we propose to use an inner product via Gadget decomposition in the relinearization based on the MK-BFV scheme instead of the original outer product operation, which can reduce the complexity of the NTT operation to (d+2d~)r′/(d+2)l~\left( {d + 2\tilde{d}} \right)r^{\prime } /\left( {d + 2} \right)\tilde{l}(d+2d~)r′/(d+2)l~ d + 2 d ~ r ′ / d + 2 l ~ of the original one. Finally, we propose a MK-BFV without CRS on the basis of the previous ones, which enhances the user's control over his own key.