Lattice-based hard problems are a leading candidate for implementation in future public key cryptographic schemes due to their conjectured quantum resilience. Lattice-based problems offer certain advantages over non-lattice-based cryptosystems, such as a relatively short key length [3] and versatility, since lattice cryptosystems can offer both encryption schemes (to securely transmit data from sender to receiver) and signature schemes (used for a receiver to verify that information actually originated from the claimed sender) [2]. Notably they are also the only known class of problems which give rise to fully homomorphic encryption schemes, in which computations can be securely performed on encrypted data [1]. Many of the 2017 submissions to the NIST Post-Quantum Cryptography challenge are based on lattice problems [4].
Lattice-based hard problems are a leading candidate for implementation in future public key cryptographic schemes due to their conjectured quantum resilience. Lattice-based problems offer certain advantages over non-lattice-based cryptosystems, such as a relatively short key length [3] and versatility, since lattice cryptosystems can offer both encryption schemes (to securely transmit data from sender to receiver) and signature schemes (used for a receiver to verify that information actually originated from the claimed sender) [2]. Notably they are also the only known class of problems which give rise to fully homomorphic encryption schemes, in which computations can be securely performed on encrypted data [1]. Many of the 2017 submissions to the NIST Post-Quantum Cryptography challenge are based on lattice problems [4].
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.
Public-key cryptography is an indispensable component used in almost all of our present-day digital infrastructure. However, most if not all of it is predominantly built upon hardness guarantees of number theoretic problems that can be broken by large-scale quantum computers in the future. Sensing the imminent threat from continued advances in quantum computing, NIST has recently initiated a global-level standardization process for quantum resistant public-key cryptographic primitives such as public-key encryption, digital signatures, and key encapsulation mechanisms. While the process received proposals from various categories of post-quantum cryptography, lattice-based cryptography features most prominently among all the submissions. Lattice-based cryptography offers a very attractive alternative to traditional public-key cryptography mainly due to the variety of lattice-based schemes offering varying flavors of security and efficiency guarantees. In this article, we survey the evolution of lattice-based key-sharing schemes (public-key encryption and key encapsulation schemes) and cover various aspects ranging from theoretical security guarantees, general algorithmic frameworks, practical implementation aspects, and physical attack security, with special focus on lattice-based key-sharing schemes competing in the NIST’s standardization process.
The imminent threat posed by the development of cryptographically relevant quantum computers necessitates a paradigm shift in public-key cryptography. Traditional cryptosystems like RSA and Elliptic Curve Cryptography (ECC), which underpin global digital security, are vulnerable to quantum algorithms such as Shor's algorithm. This vulnerability mandates a proactive migration to post-quantum cryptography (PQC) to secure long-term sensitive data. Among the various candidates for post-quantum security, lattice-based cryptography has emerged as a particularly promising and robust solution. This paper explores the fundamental principles of lattice-based cryptography, detailing its security foundations rooted in the presumed hardness of lattice problems like the Short Integer Solution (SIS) and Learning With Errors (LWE) problems. We review the significant progress in the field, including its selection as a primary candidate in the NIST Post-Quantum Cryptography standardization process. We discuss its advantages in terms of provable security, versatility, and potential for efficient implementation across diverse platforms, from cloud infrastructure to the Internet of Things (IoT). We also address the practical implications and challenges of its deployment, asserting its indispensable role in fortifying public-key security against future quantum attacks, thereby establishing it as a post-quantum imperative.
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.
Why study Lattice-based Cryptography? There are a few ways to answer this question. 1. It is useful to have cryptosystems that are based on a variety of hard computational problems so the different cryptosystems are not all vulnerable in the same way. 2. The computational aspects of lattice-based cryptosystem are usually simple to understand and fairly easy to implement in practice. 3. Lattice-based cryptosystems have lower encryption/decryption computational complexities compared to popular cryptosystems that are based on the integer factorisation or the discrete logarithm problems. 4. Lattice-based cryptosystems enjoy strong worst-case hardness security proofs based on approximate versions of known NP-hard lattice problems. 5. Lattice-based cryptosystems are believed to be good candidates for post-quantum cryptography, since there are currently no known quantum algorithms for solving lattice problems that perform significantly better than the best-known classical (non-quantum) algorithms, unlike for integer factorisation and (elliptic curve) discrete logarithm problems. 6. Last but not least, interesting structures in lattice problems have led to significant advances in Homomorphic Encryption, a new research area with wide-ranging applications.
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.