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Journal of Mathematical Cryptology

On the exact decryption range for Gentry–Halevi's implementation of fully homomorphic encryption

Masaya Yasuda, Kazuhiro Yokoyama, Takeshi Shimoyama, Jun Kogure, Takeshi Koshiba

Published Mar 12, 2014• 3 citations
Publisher

Abstract

Abstract In this paper, we revisit the fully homomorphic encryption (FHE) scheme implemented by Gentry and Halevi, which is just an instantiation of Gentry's original scheme based on ideal lattices. Their FHE scheme starts from a somewhat homomorphic encryption (SHE) scheme, and its decryption range is deeply related with the FHE construction. Gentry and Halevi gave an experimental evaluation of the decryption range, but theoretical evaluations have not been given so far. Moreover, we give a theoretical upper bound, and reconsider suitable parameters for theoretically obtaining an FHE scheme. In particular, while Gentry and Halevi use the Euclidean norm evaluation in the noise management of ciphertexts, our theoretical bound enables us to use the ∞-norm evaluation, and hence it helps to lower the difficulty of controlling the noise density of ciphertexts.

DOI
10.1515/jmc-2013-0024

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Abstract In this paper, we revisit the fully homomorphic encryption (FHE) scheme implemented by Gentry and Halevi, which is just an instantiation of Gentry's original scheme based on ideal lattices. Their FHE scheme starts from a somewhat homomorphic encryption (SHE) scheme, and its decryption range is deeply related with the FHE construction. Gentry and Halevi gave an experimental evaluation of the decryption range, but theoretical evaluations have not been given so far. Moreover, we give a theoretical upper bound, and reconsider suitable parameters for theoretically obtaining an FHE scheme. In particular, while Gentry and Halevi use the Euclidean norm evaluation in the noise management of ciphertexts, our theoretical bound enables us to use the ∞-norm evaluation, and hence it helps to lower the difficulty of controlling the noise density of ciphertexts.
A critical challenge in a fully homomorphic encryption (FHE) scheme is to manage noise. Modulus switching technique is currently the most efficient noise management technique. When using the modulus switching technique to design and implement a FHE scheme, how to choose concrete parameters is an important step, but to our best knowledge, this step has drawn very little attention to the existing FHE researches in the literature. The contributions of this paper are twofold. On one hand, we propose a function of the lower bound of dimension value in the switching techniques depending on the LWE specific security levels. On the other hand, as a case study, we modify the Brakerski FHE scheme (in Crypto 2012) by using the modulus switching technique. We recommend concrete parameter values of our proposed scheme and provide security analysis. Our result shows that the modified FHE scheme is more efficient than the original Brakerski scheme in the same security level.
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.
This paper proposes a generic construction of CCA-secure fully homomorphic encryption (FHE) scheme.First, we introduce a new primitive called multi-identity fully homomorphic encryption, which works in the multi-identity setting; that is, homomorphic evaluation can performed on ciphertexts created with different identities.Then, we propose an IND-sID-CPA secure multi-identity leveled FHE scheme over lattice.Finally, we present CCA-secure FHE from our proposed multi-identity FHE scheme and strongly EUF-CMA secure signature.The security analysis shows that our FHE scheme is CCA2-secure when the evaluation key is unavailable to the adversary, and remains CCA1-secure when the evaluation key is exposed in the standard model.
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.
The current multikey fully homomorphic encryption (MKFHE) needs to add exponential noise in the distributed decryption phase to ensure the simulatability of partial decryption. Such a large noise causes the ciphertext modulus of the scheme to increase exponentially compared to the single‐key fully homomorphic encryption (FHE), further reducing the efficiency of the scheme and making the hardness problem on the lattice on which the scheme relies have a subexponential approximation factor (which means that the security of the scheme is reduced). To address this problem, this paper analyzes in detail the noise in partial decryption of the MKFHE based on the learning with error (LWE) problem. It points out that this part of the noise is composed of private key and the noise in initial ciphertext. Therefore, as long as the encryption scheme is leak‐resistant and the noise in partial decryption is independent of the noise in the initial ciphertext, the semantic security of the ciphertext can be guaranteed. In order to make the noise in the initial ciphertext independent of the noise in the partial decryption, this paper proves the smudging lemma on discrete Gaussian distribution and achieves this goal by multiplying the initial ciphertext by a “dummy” ciphertext with a plaintext of 1. Based on the above method, this paper removes the exponential noise in the distributed decryption phase for the first time and reduces the ciphertext modulus of MKFHE from 2 ω ( λ L log λ ) to 2 O ( λ + L ) as the same level as the FHE.
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.