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.
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.
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.
As quantum computing technology continues to advance, the development of post-quantum cryptography (PQC) methods capable of resisting quantum computing attacks has emerged as a key research direction. Existing homomorphic encryption schemes, such as RSA, ElGamal, and Paillier, face potential vulnerabilities to quantum attacks. To address this challenge, this study proposes a lattice-based PQC homomorphic encryption scheme. The approach leverages lattice cryptographic methods to resist quantum attacks while enabling homomorphic encryption for computational applications. This research provides mathematical proofs, computational examples, and a comprehensive security analysis of the proposed lattice-based PQC homomorphic encryption scheme. The findings aim to serve as a reference for developers working on homomorphic encryption applications, such as federated learning.
Abstract The rapid evolution of quantum computing fundamentally threatens modern cryptographic infrastructures, necessitating a paradigm shift toward advanced encryption techniques. While post-quantum cryptography (PQC) ensures that data remains secure against quantum algorithmic decryption, it does not inherently facilitate the processing of that data in untrusted environments. Homomorphic encryption (HE) addresses this computational privacy requirement, yet traditional HE schemes rely on classical mathematical problems that are easily compromised by quantum systems. This paper presents a comprehensive comparative framework that integrates post-quantum cryptographic principles with homomorphic capabilities to enable quantum-resistant, privacy-preserving computation. By analyzing lattice-based algorithms, code-based cryptography, and hardware acceleration paradigms, this study maps the existing trade-offs between security, computational overhead, and practical deployment feasibility in the impending quantum era. Keywords: PQC, HE. 1.Introduction The rapid evolution of quantum computing technology presents a profound threat to modern cryptographic infrastructures. As quantum algorithms, notably Shor's algorithm, mature, they acquire the capability to solve nondeterministic polynomial time problems in polynomial time, thereby breaking traditional asymmetric encryption standards such as RSA and Elliptic Curve Cryptography (ECC) (Chen, 2024). Consequently, the development of post-quantum cryptography (PQC) has become a global imperative to secure sensitive communications and data against future quantum adversaries (Pranjal & Chaturvedi, 2024). Cryptographers are currently focusing on transitioning secure systems to these newly established PQC standards before large-scale quantum computers become commercially viable. While PQC addresses the secure transmission and storage of data, it does not inherently solve the problem of secure data processing in untrusted cloud environments. Homomorphic encryption (HE) provides a sophisticated mechanism that allows computations to be performed directly on encrypted data without requiring prior decryption (Jain & Cherukuri, 2023). The core problem addressed in this paper is the conceptual and practical integration of PQC with HE shown in Fig1.1 which is essential for ensuring quantum-safe, privacy-preserving computation in distributed architectures. Achieving this synthesis is highly complex due to the massive computational overhead and noise accumulation inherent to both cryptographic domains. However, existing approaches to privacy-preserving computation remain broadly insufficient for the post-quantum era. First, conventional homomorphic encryption schemes rely heavily on mathematical hardness assumptions, such as integer factorization or discrete logarithms, which are demonstrably vulnerable to quantum attacks (Chen, 2024). Second, while some high-performance fully homomorphic implementations exist, they suffer from prohibitive computational overhead and severe ciphertext expansion, making widespread deployment impractical without specialized mitigation strategies (Tseng et al., 2025). Third, hardware-accelerated privacy solutions often rely on trusted execution environments, like Intel SGX, which inherently introduce side-channel vulnerabilities and force reliance on centralized hardware trust rather than pure mathematical security (Sadat et al., 2017). To overcome these critical limitations, this paper proposes a structured comparative methodology for evaluating quantum-resistant homomorphic encryption systems. Specifically, our paper makes the following main contributions: We provide a comprehensive comparative framework that categorizes and evaluates lattice-based and code-based homomorphic encryption schemes against classical performance constraints. We propose a structured, hypothetical evaluation pipeline designed to benchmark the computational overhead and latency of post-quantum homomorphic algorithms in federated networking environments. 2.Privacy-preserving computation The literature surrounding privacy-preserving computation can be broadly classified into three primary categories, the first being classical and fully homomorphic encryption (FHE) systems. The core idea behind these schemes is the preservation of algebraic structures during encryption, allowing operations on ciphertexts to reflect directly on the underlying plaintexts without exposing the raw data (Jain & Cherukuri, 2023). A major strength of these classical FHE systems, such as those utilized in encrypted relational databases like FHE-SQL, is their ability to provide end-to-end cryptographic protection for queries without relying on trusted hardware modules (Tseng et al., 2025). However, a fundamental weakness is that early FHE models, and their underlying privacy homomorphisms, remain fundamentally susceptible to quantum cryptanalysis due to their reliance on classical mathematical problems (Sen, 2013). Compared to these classical paradigms, our work specifically focuses on migrating these operational homomorphic properties into genuinely quantum-safe mathematical frameworks. The second category of related work focuses on hardware acceleration and hybrid computational environments for homomorphic encryption. The core idea here involves leveraging specialized hardware, such as ASIC AI chips or coarse-grained compute architectures like Google's TPUs, to mitigate the enormous computational overhead associated with HE (Tong et al., 2025). The strength of this approach lies in its ability to bridge the energy-efficiency gap, often utilizing intelligent compiler frameworks to convert high-precision modular arithmetic into dense, low-precision matrix multiplications. Conversely, the weakness of hardware-centric methods is that architectural mismatches frequently lead to severe resource under-utilization, and hybrid solutions utilizing components like Intel SGX introduce hardware trust assumptions that can compromise pure cryptographic integrity (Sadat et al., 2017). Unlike these approaches, our framework prioritizes algorithmic resilience and quantum resistance over purely hardware-driven and enclave-based optimizations. The third category encompasses emerging post-quantum and quantum homomorphic paradigms, which seek to unify quantum resistance with computational privacy. Researchers have proposed lattice-based HE schemes to replace vulnerable legacy systems like Paillier and ElGamal, successfully enabling secure applications such as federated learning (Chen, 2024). Similarly, code-based cryptography utilizing McEliece and non-negative matrix factorization has shown promise in providing faster encryption and decryption times while remaining mathematically secure against quantum computers (Chen, 2024). Furthermore, advanced quantum homomorphic encryption protocols have explored information-theoretic security combined with homomorphic quantum error-correction to ensure future-proof data protection (Ouyang & Rohde, 2022). While these systems represent the state-of-the-art, their primary weakness remains the massive memory and communication overhead introduced during distributed training protocols, which negatively impacts classification performance. Our approach builds upon this category by proposing a standardized methodology to systematically compare these disparate post-quantum homomorphic techniques. 3.Multi-stage Method To systematically evaluate and deploy post-quantum homomorphic encryption, we propose a structured, multi-stage methodological framework. The first step involves cryptographic primitive selection, where system architects must evaluate and choose between lattice-based constructs, such as ML-KEM adaptations, and code-based systems like McEliece, depending on their domain constraints (Chen, 2024). The second step requires the implementation of an algorithmic parameter optimization module, which dynamically adjusts polynomial degrees and error distributions to balance security levels with the operational multiplicative depth required by the application. Finally, the third step comprises the integration of a specialized abstraction layer designed to map complex high-precision cryptographic operations into hardware-friendly matrix operations, thereby reducing the inherent bottleneck of ciphertext evaluation. The key design choices within this framework are driven by the absolute necessity to balance quantum resistance with practical algorithmic utility. We prioritize lattice-based cryptography for our primary evaluation pathway because of its established resilience against quantum threats and its recent standardization by the National Institute of Standards and Technology (NIST) (Chen, 2024). Additionally, we explicitly design our framework to avoid reliance on trusted hardware environments, opting instead for pure cryptographic guarantees to prevent the side-channel data leaks that often plague hardware-enclave solutions. By focusing on algorithmic alignment and basis-aligned mathematical transformations, the framework ensures that homomorphic workloads can be optimized effectively without compromising the underlying post-quantum security proofs. To validate the efficacy of the proposed framework, we outline a comprehensive evaluation plan utilizing a hypothetical, high-dimensional healthcare benchmarking dataset. The experimental setup will simulate a secure genome-wide association study (GWAS) executing within a fully decentralized federated learning environment (Sadat et al., 2017). The primary metrics for this hypothetical evaluation will include key generation latency, ciphertext expansion ratio, and the amortized computational time required for homomorphic multiplication operations across distributed nodes. By systematically varying the cryptographic noise parameters and observing the resulting degradation in analytical performance, we aim to empirically map the trad
Classical cryptography has been around for a long time in the documented human history, but most classical ciphers were broken and even solved by hand. Shannon introduced the notion of perfect secrecy that formally defines confidentiality in the information-theoretic sense, which is only possible in the restricted scenarios where the message is no longer than the encryption key. The invention of public-key cryptography (the Diffie-Hellman key exchange protocol in 1976 and the RSA crypto-system in 1977) marks the birth of modern cryptography, allowing parties to exchange messages securely without sharing any secrets in advance. Furthermore, it provides computational security based on the conjectured hardness of mathematical problems such as factorization and the discrete logarithm. Public-key cryptography has found numerous applications in the Internet, financial and banking industry, and blockchains, and it plays a crucial role in protecting information security and asset safety. Unfortunately, in the 1990s, Shor proposed efficient quantum algorithms that solve number-theoretic problems, including factorization and discrete logarithms in polynomial time. Once a quantum computer of a particular scale becomes a reality, it will cause a devastating blow to the existing public-key infrastructure. To deal with such a ‘quantum crisis’, academia and industry are looking into the design, analysis and standardization of cryptographic algorithms that can resist quantum computers referred to as post-quantum cryptography (PQC). The National Institute of Standards and Technology (NIST) has been soliciting proposals for the post-quantum public-key algorithms since 2016. More recently, the Chinese Association for Cryptologic Research (CACR) held a competition on designing cryptographic algorithms whose public-key cryptography track focused on post-quantum cryptographic algorithms. Lattice-based cryptography is considered by most to be the mainstream technical route of post-quantum cryptography, which is reflected in the number of proposals (and their percentage of the total) received in the NIST PQC process. To reflect the status quo of post-quantum cryptography, we invite leading experts in this area to contribute three technical perspectives that aim to help readers understand the algorithms, the underlying basic techniques and different technical routes to achieve quantum resistance. The first perspective, presented by Lu and Zhang, introduces public-key cryptographic algorithms whose quantum security is reducible from the conjectured quantum hardness of lattice problems. In particular, they mainly focus on public-key encryption (PKE) and the key encapsulation mechanism (KEM), which are essential building blocks for securing the confidentiality of communication without pre-shared secrets. Both types of crypto-systems are solicited by the NIST PQC standardization and the CACR algorithm design competition. This perspective gives a comprehensive survey on practical lattice-based PKEs/KEMs, and their best-known quantum and classical attacks. Another important post-quantum crypto-system is digital signature, which ensures that three goals of information security are met other than confidentiality, namely, integrity, authentication and non-repudiation. The second perspective is on lattice-based signature by Lyubashevsky. In this perspective, he surveys different techniques in building lattice-based post-quantum crypto-systems, discusses the challenges in overcoming performance issues and gives us state-of-the-art digital signature schemes. In addition to ensuring the ‘static’ security of information in storage and transmission, advanced cryptographic algorithms and protocols can guarantee information security during the computation process (possibly among multiple parties), referred to as privacy-preserving computation. Cryptographic techniques involved in privacy-preserving computation include secure multi-party computation, zero-knowledge proof and fully homomorphic encryption. There is a pressing need to migrate them to the post-quantum era. The third perspective, by Yu and Xie, presents practical instantiations of these algorithms and discusses possible ways to migrate them to their quantum-resistant counterparts. To summarize, post-quantum cryptography has received widespread attention and made significant progress in recent years. Some post-quantum cryptographic algorithms, such as the lattice-based candidate, also have other advantages (e.g., computational efficiency and full homomorphism) over their classical counterparts. Lattice-based cryptography is an emerging field with high theoretical value and wide application, and we encourage young researchers to enter and explore this new and exciting field.
Lattice cryptography has become one of the focal points in the era of quantum computing development. RSA and ECC, traditional public key cryptosystems, may become insecure in the face of quantum computers. In contrast, lattice cryptography is considered a potential solution for cryptographic security in the quantum computing age due to its inherent mathematical problems that require the use of quantum computers to solve. Moreover, lattice cryptographic algorithms can achieve efficient encryption and decryption processes and support a variety of cryptographic constructions, including encryption, signatures, and fully homomorphic encryption (FHE). With its high efficiency, multifunctionality, and high security, lattice cryptography has applications in various fields such as finance, e-commerce, government agencies, military, and network communications. It has also become one of the top contenders for the post-quantum cryptographic algorithm standards that have been recognized by the National Institute of Standards and Technology (NIST). This paper discusses the main contributions of lattice cryptography to society and its potential future development directions, as well as the challenges it may face and potential solutions. This thesis elaborates on lattice-based cryptography and comprehensively discusses the key technologies of lattice cryptography, its applications, security analysis, performance evaluation, and the latest progress.