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IEEE International Conference on Trust, Security and Privacy in Computing and Communications (TrustCom)

New Compact Construction of FHE from Cyclic Algebra LWE

Yuan Liu, Licheng Wang, Yongbin Zhou

Published Dec 17, 2024• 0 citations

Abstract

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.

DOI: 10.1109/trustcom63139.2024.00102

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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: In this paper, we first introduce a new concept of approximate lattice problem (ALP), which is an extension of learning with errors (LWE). Next, we propose two ALP-based public key encryption schemes. Then, we construct two new fully homomorphic encryption scheme (FHE) based on respectively approximate principal ideal lattice problem with related modulus (APIP-RM) and approximate lattice problem with related modulus (ALP-RM). Moreover, we also extend our ALP-RM-based FHE to the ALP problem with unrelated modulus (ALP-UM). Our work is different from previous works in three aspects: (1) We extend the LWE problem to the ALP problem. This ALP problem is similar to the closest vector problem in lattice. We believe that this problem is independent of interest. (2) We construct a new FHE by using a re-randomizing method, which is different from the squashing decryption in previous works. (3) The expansion rate is merely O(k) with k a security parameter in Our FHE, which can be improved to O(logk) by using dimension reduction [BV11], whereas all previous schemes are at least O(k*logk) [BV11, Gen11, LNV11]. Our method can also decrease a factor k of the expansion rate in their schemes.

In this paper, we first introduce a new concept of approximate lattice problem (ALP), which is an extension of learning with errors (LWE). Next, we propose two ALP-based public key encryption schemes. Then, we construct two new fully homomorphic encryption scheme (FHE) based on respectively approximate principal ideal lattice problem with related modulus (APIP-RM) and approximate lattice problem with related modulus (ALP-RM). Moreover, we also extend our ALP-RM-based FHE to the ALP problem with unrelated modulus (ALP-UM). Our work is different from previous works in three aspects: (1) We extend the LWE problem to the ALP problem. This ALP problem is similar to the closest vector problem in lattice. We believe that this problem is independent of interest. (2) We construct a new FHE by using a re-randomizing method, which is different from the squashing decryption in previous works. (3) The expansion rate is merely O(k) with k a security parameter in Our FHE, which can be improved to O(logk) by using dimension reduction [BV11], whereas all previous schemes are at least O(k*logk) [BV11, Gen11, LNV11]. Our method can also decrease a factor k of the expansion rate in their schemes.

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) 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) 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.

Very recently, Costache and Smart proposed a fully homomorphic encryption (FHE) scheme based on the Learning with Rounding (LWR) problem, which removes the noise (typically, Gaussian noise) sampling needed in the previous lattices-based FHEs. But their scheme did not work, since the noise of homomorphic multiplication is complicated and large, which leads to failure of decryption. More specifically, they chose LWR instances as a public key and the private key therein as a secret key and then used the tensor product to implement homomorphic multiplication, which resulted in a tangly modulus problem. Recall that there are two moduli in the LWR instances, and then the moduli will tangle together due to the tensor product. Inspired by their work, we built the first workable LWR-based FHE scheme eliminating the tangly modulus problem by cleverly adopting the celebrated approximate eigenvector method proposed by Gentry et al. at Crypto 2013. Roughly speaking, we use a specific matrix multiplication to perform the homomorphic multiplication, hence no tangly modulus problem. Furthermore, we also extend the LWR-based FHE scheme to the multikey setting using the tricks used to construct LWE-based multikey FHE by Mukherjee and Wichs at Eurocrypt 2016. Our LWR-based multikey FHE construction provides an alternative to the existing multikey FHEs and can also be applied to multiparty computation with higher efficiency.