Lattice-based cryptography is encryption that uses lattice issues as the basis of a conjecture of the complexity of the worst-case problem. This complexity is considered hard in the world of classical and quantum providing for a primitives design of lattice-based cryptography algorithms. The NIST held a contest to find the best algorithm to implement. This paper compares three finalists of that contest by analyzing the theoretical basis of the lattice problems they hold, with the security of the worst case of the attack through some algorithms, lattices with underlying problems, construction of cryptographic algorithms, and comparative analysis of three finalists of the lattice schemes of the NIST competition. Comparison includes the safety and robustness of the lattice problem used the cryptographic algorithm that is built and finally there will be an analysis in a computational time context to discuss the efficiency algorithm created. Two advantages and disadvantages of each the algorithm finalist are implemented and also described.
Lattice-based cryptography is encryption that uses lattice issues as the basis of a conjecture of the complexity of the worst-case problem. This complexity is considered hard in the world of classical and quantum providing for a primitives design of lattice-based cryptography algorithms. The NIST held a contest to find the best algorithm to implement. This paper compares three finalists of that contest by analyzing the theoretical basis of the lattice problems they hold, with the security of the worst case of the attack through some algorithms, lattices with underlying problems, construction of cryptographic algorithms, and comparative analysis of three finalists of the lattice schemes of the NIST competition. Comparison includes the safety and robustness of the lattice problem used the cryptographic algorithm that is built and finally there will be an analysis in a computational time context to discuss the efficiency algorithm created. Two advantages and disadvantages of each the algorithm finalist are implemented and also described.
The need and emergence of quantum computing under National Quantum Mission threatens the cyber-security of existing cryptographic protocols, especially public-key schemes built on integer factorization and discrete logarithms. Lattice-based cryptography has gained prominence as a robust candidate for post-quantum cryptography—offering security against quantum attacks while enabling efficient operations, parallelization, and novel functionalities. This paper analyzes the mathematical foundations of lattices, explores hardness assumptions underlying widely studied lattice problems, highlights prominent cryptographic constructions, analyzes performance trade-offs, and reviews standardization efforts. This paper concludes by identifying ongoing issues and future directions for lattice-based cryptographic systems in a quantum-secure digital world. The algorithm has been implemented in Python.
Post-quantum cryptography (PQC) has emerged as a critical research area for securing communications in the quantum era. Among the selected algorithms, lattice-based schemes have gained particular attention due to their strong security foundations and practical efficiency.This paper presents a comparative analysis of latticebased PQC algorithms from the U.S. National Institute of Standards and Technology (NIST) standardization process and the Korean PQC (KpqC) competition. The evaluation considers security levels, key and ciphertext/signature sizes, and implementation performance. The results highlight fundamental trade-offs between security, efficiency, compactness, and implementation complexity, providing insights into the strengths and limitations of current latticebased algorithms. These findings offer practical guidance for assessing their suitability for deployment in real-world post-quantum systems.
The security of classical public-key cryptosystems, which operate based on the computational intractability of integer factorization and discrete logarithm problems, is profoundly threatened by the rapid development of quantum computing. Post-Quantum Cryptography (PQC) has been proposed to improve security against quantum attacks. Lattice-based cryptography is widely regarded as one of the most promising PQC method because of its proven security, efficient implementation, and functional versatility, and it has achieved significant progress in both algorithmic design and engineering implementation in recent years. This paper presents a systematic survey of the status of research on lattice-based cryptographic algorithms. First, the fundamental theory of lattices and their core hard problems are discussed. Next, the technical characteristics and performance of encryption and key encapsulation mechanisms, digital signature schemes, and key exchange protocols constructed based on lattice hard problems are reviewed and compared. Subsequently, the principles and optimization paths of two key technologies for lattice cryptography implementation, namely polynomial multiplication using the fast Number Theoretic Transform (NTT) and pre-image sampling algorithms, are discussed. Finally, considering the PQC standardization process of the National Institute of Standards and Technology (NIST), an outlook on future research directions for lattice-based cryptography is presented. The research directions include the development of more "succinct" fundamental hard problems, strengthening security proofs, improving algorithm efficiency, and countering novel attacks. This review serves as a reference for promoting the practical deployment and further research of lattice-based cryptography.
Alongside the growth of technology is the increasing demand for security of data and communication channels. As attacking power grows, devices become smaller and data expands, the security needs to be stronger yet less resource-consuming than ever before. Public key cryptography (PKC) is used in the majority of today's communications systems. The security of public-key cryptographic schemes, such as RSA and elliptic curve cryptography (ECC), is based on underlying mathematical problems such as factoring large integers or solving the discrete log problem. However, the discovery of Shor's Algorithm in 1994 insinuates that these schemes will be insecure when a full-scale quantum computer is developed, as these problems will be solvable in polynomial time. Therefore attention has moved towards post-quantum, or quantum-safe, cryptography (QSC). In 2016, the National Institute of Standards and Technology (NIST) announced the beginning of their efforts to migrate to QSC, a standardisation process which has involved the cryptographic community worldwide. There are five main types of QSC being considered for standardisation; lattice-based, code-based, multivariate-based, hash-based and isogeny-based. Lattice-based cryptography (LBC) offers a promising alternative to today's classical approaches. Hard problems based on lattices are believed to be unsolvable, even by a quantum-computer, in a feasible amount of time. In 1995, Ajtai showed that the average case complexity of these problems is at least as hard as solving the problems in the worst case. Furthermore, lattice-based schemes are the only remaining contender in the NIST process to provide the important primitives of encryption, key encapsulation and digital signature schemes (DSSs). A well-recognised open problem surrounding quantum-safe cryptography is its increase in bandwidth and computational complexity in comparison to existing cryptography. However, in comparison to other quantum-safe schemes, LBC has been shown to be relatively efficient. In this thesis, novel software designs of lattice-based schemes are presented and shown to be competitive with existing PKC. Additionally, physical attacks are considered and countermeasures are incorporated into the designs. DSSs are required for authentication, integrity and non-repudiation of data. This thesis provides an evaluation of three leading signature schemes, BLISS-B, Dilithium and \textsc{Falcon}, which vary in terms of their key features, such as the sampling processes used, and the structure of the lattices. Physical attacks on the three schemes are considered and the effect on performance of appropriate attack countermeasures is evaluated. A novel fault attack on the NIST candidate \textsc{Falcon} is also proposed. A further advantage of LBC is the ability to support cryptographic primitives with extended capabilities to facilitate certificate management and compute on encrypted data. The complexity of today's technology advances mean applications require more sophisticated primitives, which contrastingly are required to run on even smaller platforms. This thesis presents novel quantum-safe advanced cryptographic schemes, specifically lattice-based identity-based encryption (IBE) and hierarchical identity-based encryption (HIBE). IBE removes the need for certificate management, as a user's public key depends on their public identity. IBE is currently used in vital applications such as emergency services communication and so the development of a high-performance quantum-safe alternative is imperative. This thesis also introduces the first scalable HIBE scheme based on lattices. The first HIBE scheme based on structured lattices, Latte, was proposed by the National Centre for Cyber Security (NCSC) in 2017. However, the public key and ciphertext sizes increased with each delegation and hence it was not feasible to scale the algorithm. Skinny Latte, proposed in this thesis, remedies this issue by fixing the dimension of the lattice, thereby making public key and ciphertext sizes constant. This research also shows that it is practical for real-world deployment. Overall, this thesis demonstrates that lattice-based cryptography is practical for real-world applications. It considers both theoretical and physical attacks, and presents novel designs of advanced quantum-safe cryptographic primitives, which have only yet been realised by lattices.
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.
Quantum cryptography is a branch of cryptography that is a mixture of quantum mechanics and classical cryptography. The study of quantum cryptography is to design cryptographic algorithms and protocols that are against quantum computing attacks. In this chapter, the authors focus on analyzing characteristics of the quantum-proof cryptosystem and its applications in the future internet. Lattice-based cryptography provides a much stronger belief of security, in that the average-case of certain problems is equivalent to the worst-case of those problems. With the increase in cryptanalytic attacks conventional cryptographic schemes will soon become obsolete. As the reality of quantum computing approaches, these cryptosystems will need to be replaced with efficient quantum-resistant cryptosystems. We need an alternate security mechanism which is as hard as the existing number theoretic approaches. In this chapter, the authors discuss the security dimension of lattice-based cryptography whose strength lies in the hardness of lattice problems and also study its application areas.