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International Symposium on AI and Cybersecurity (ISAICS)

A Research on the Application Comparison of NTRU in Typical Fully Homomorphic Encryption Scheme

Xu Zhao, Zheng Yuan

Published Oct 24, 2025• 0 citations
Publisher

Abstract

Fully homomorphic encryption (FHE) allows arbitrary computations to be performed directly on encrypted data and is a core technology for privacy computing and secure cloud computing. The Number Theory Research Unit (NTRU) has become a research topic in this field by virtue of its lattice-based anti-quantum security, efficient polynomial operation characteristics and natural multi-key homomorphism capability. In this paper, we systematically review the key advances of NTRU-based FHE, covering the directions of the design of FHE, the optimization of bootstrapping techniques, and the application to FHE for multi-key approaches. Furthermore, the innovativeness and practical application challenges of the schemes are analyzed, and the future research trends are envisioned.

DOI
10.1109/isaics66888.2025.11349745

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Fully homomorphic encryption (FHE) allows arbitrary computations to be performed directly on encrypted data and is a core technology for privacy computing and secure cloud computing. The Number Theory Research Unit (NTRU) has become a research topic in this field by virtue of its lattice-based anti-quantum security, efficient polynomial operation characteristics and natural multi-key homomorphism capability. In this paper, we systematically review the key advances of NTRU-based FHE, covering the directions of the design of FHE, the optimization of bootstrapping techniques, and the application to FHE for multi-key approaches. Furthermore, the innovativeness and practical application challenges of the schemes are analyzed, and the future research trends are envisioned.
Multi-key fully homomorphic encryption (MKFHE), a generalization of fully homomorphic encryption (FHE), enables a computation over encrypted data under multiple keys. The first MKFHE schemes were based on the NTRU primitive, however these early NTRU based FHE schemes were found to be insecure due to the problem of over-stretched parameters. Recently, in the case of standard (non-multi key) FHE a secure version, called FINAL, of NTRU has been found. In this work we extend FINAL to an MKFHE scheme, this allows us to benefit from some of the performance advantages provided by NTRU based primitives. Thus, our scheme provides competitive performance against current state-of-the-art multi-key TFHE, in particular reducing the computational complexity from quadratic to linear in the number of keys.
With the popularization of cloud computing model, the outsourcing of data storage and computing services has become an indispensable trend, which lead to the related data security and privacy protection issues that have attracted extensive attention in the industry. Fully homomorphic encryption, as an encryption technology that can process ciphertext information without exposing plaintext information, has natural user data security and privacy protection characteristics. Meanwhile, the excellent quantum-resistant performance and homomorphic properties of lattice ciphers have made lattice-based homomorphic encryption schemes a much-attend research hotspot in the field of cryptography in recent years. In this paper we mainly introduce the status of all-pass encryption and several typical references for the research of lattice-based all-homomorphic cryptography.
Since Gentry's breakthrough construction of fully homomorphic encryption from lattice-based assumptions (STOC 2009), homomorphic cryptography has attracted a lot of attention. In short, homomorphic cryptography schemes allow performing computation on encrypted data without knowing anything about the underlying plaintext. This branch of cryptography has become increasingly useful in building new protocols and schemes with intriguing security and functionality features. In this thesis, we continue to study the applications of homomorphic cryptography and the lattice-based techniques underlying them in realizing new and enhanced cryptographic primitives. We obtain the following results: -We construct the first noninteractive zero knowledge argument (and proof) system for all of NP from standard lattice assumptions. Noninteractive zero knowledge argument systems have found many applications in enhancing the functionality as well as the security of cryptographic schemes and protocols. Constructing noninteractive zero knowledge arguments from lattice assumption has been a long standing open question. We finally close this problem. -We consider multi-key fully homomorphic encryption (FHE) schemes. Traditional fully homomorphic encryption schemes allow computation on plaintext encrypted under a single key. The notion of multi-key fully homomorphic encryption allows homomorphic computation on data encrypted under different keys. We construct multi-key FHE schemes which are naturally dynamic: ciphertexts under new keys can join even during the homomorphic computation. -Finally, we focus on constrained pseudorandom functions (C-PRFs), which are pseudorandom functions (PRFs) with additional functional capabilities. We propose a new approach for building C-PRFs from lattices, and also significantly enhance the underlying lattice parameters.
Abstract. Fully homomorphic encryption (FHE) is a form of public-key encryption that enables arbitrary computation over encrypted data. The past few years have seen several realizations of FHE under differ-ent assumptions, and FHE has been used as a building block in many cryptographic applications. Adaptive security for public-key encryption schemes is an important se-curity notion proposed by Canetti et al. It is intended to ensure security when encryption is used within an interactive protocol and parties may be adaptively corrupted by an adversary during the course of the protocol execution. Due to the extensive applications of FHE to protocol design, it is natural to understand whether adaptively secure FHE is achievable. In this paper we show two contrasting results in this direction. First, we show that adaptive security is impossible for FHE satisfying the (stan-dard) compactness requirement. On the other hand, we show a construc-tion of adaptively secure FHE that is not compact, but that does achieve circuit privacy.
The Multi-Key Fully Homomorphic Encryption (MKFHE) based on the NTRU cryptosystem is an important alternative to the post-quantum cryptography due to its simple scheme form, high efficiency, and fewer ciphertexts and keys. In 2012, Lopez-Alt et al. proposed the first NTRU-type MKFHE scheme, the LTV12 scheme, using the key-switching and modulus-reduction techniques, whose security relies on two assumptions: the Ring Learning With Error (RLWE) assumption and the Decisional Small Polynomial Ratio (DSPR) assumption. However, the LTV12 and subsequent NTRU-type schemes are restricted to the family of power-of-2 cyclotomic rings, which may affect the security in the case of subfield attacks. Moreover, the key-switching technique of the LTV12 scheme requires a circular application of evaluation keys, which causes rapid growth of the error and thus affects the circuit depth. In this paper, an NTRU-type MKFHE scheme over prime cyclotomic rings without key-switching is proposed, which has the potential to resist the subfield attack and decrease the error exponentially during the homomorphic evaluating process. First, based on the RLWE and DSPR assumptions over the prime cyclotomic rings, a detailed analysis of the factors affecting the error during the homomorphic evaluations in the LTV12 scheme is provided. Next, a Low Bit Discarded & Dimension Expansion of Ciphertexts (LBD&DEC) technique is proposed, and the inherent homomorphic multiplication decryption structure of the NTRU is proposed, which can eliminate the key-switching operation in the LTV12 scheme. Finally, a leveled NTRU-type MKFHE scheme is developed using the LBD&DEC and modulus-reduction techniques. The analysis shows that the proposed scheme compared to the LTV12 scheme can decrease the magnitude of the error exponentially and minimize the dimension of ciphertexts.