Lattice-based cryptography is one of the most promising candidates for designing post-quantum cryptographic algorithms that resist emerging quantum computing attacks. The recent NIST PQC standardization process is nearing its completion, with practical lattice-based algorithms for basic cryptographic functionalities (namely digital signature and public-key encryption) selected for standardization in the near future. However, practical lattice-based solutions for more advanced privacy-preserving protocols, in particular, Zero-Knowledge Proofs (ZKPs), have only emerged recently and are an active area of research. We discuss some recent developments in design and analysis of practical lattice-based post-quantum ZKPs and their applications. In particular, we review some challenges that arise in designing ZKPs in the lattice setting and some recent progress on efficient lattice-based Schnorr-like proofs for important relations, such as binary/range proofs, one-out-of-many proofs and rounding proofs [1, 2, 4]. We discuss applications and optimization of such proof systems as building blocks for practical advanced cryptographic protocols such as ring signatures and balance proofs for privacy-preserving cryptocurrency payment protocols [2, 3]. We also discuss our recent work on succinct designated-verifier ZKPs (DV-ZKSNARKS) for verifying correctness of general delegated computations [5].
Lattice-based cryptography is one of the most promising candidates for designing post-quantum cryptographic algorithms that resist emerging quantum computing attacks. The recent NIST PQC standardization process is nearing its completion, with practical lattice-based algorithms for basic cryptographic functionalities (namely digital signature and public-key encryption) selected for standardization in the near future. However, practical lattice-based solutions for more advanced privacy-preserving protocols, in particular, Zero-Knowledge Proofs (ZKPs), have only emerged recently and are an active area of research. We discuss some recent developments in design and analysis of practical lattice-based post-quantum ZKPs and their applications. In particular, we review some challenges that arise in designing ZKPs in the lattice setting and some recent progress on efficient lattice-based Schnorr-like proofs for important relations, such as binary/range proofs, one-out-of-many proofs and rounding proofs [1, 2, 4]. We discuss applications and optimization of such proof systems as building blocks for practical advanced cryptographic protocols such as ring signatures and balance proofs for privacy-preserving cryptocurrency payment protocols [2, 3]. We also discuss our recent work on succinct designated-verifier ZKPs (DV-ZKSNARKS) for verifying correctness of general delegated computations [5].
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.
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.
The hardness of lattice problems offers one of the most promising security foundations for quantum-safe cryptography. Basic schemes for public key encryption and digital signatures are already close to standardization at NIST and several other standardization bodies, and the research frontier has moved on to building primitives with more advanced privacy features. At the core of many such primitives are zero-knowledge proofs. In recent years, zero-knowledge proofs for (and using) lattice relations have seen a dramatic jump in efficiency and they currently provide arguably the shortest, and most computationally efficient, quantum-safe proofs for many scenarios. The main difficulty in using these proofs by non-experts (and experts!) is that they have a lot of moving parts and a lot of internal parameters depend on the particular instance that one is trying to prove.
This paper delves into lattice-based cryptography as a cornerstone of post-quantum cryptography (PQC), exploring its evolution, key principles, and applications. It examines how lattice-based schemes leverage the inherent hardness of problems such as the Shortest Vector Problem (SVP) and Learning With Errors (LWE) to provide quantum-resistant encryption, digital signatures, and homomorphic encryption. The paper also outlines the historical development of lattice-based cryptography, emphasizing its resilience to quantum attacks in contrast to traditional schemes like RSA and ECC, which are vulnerable to Shor's algorithm. Core concepts, including lattice trapdoors, parameter optimization, and real-world applications in blockchain, cryptocurrencies, and secure communication protocols, are explored in detail. Furthermore, the paper highlights ongoing standardization efforts led by NIST, ethical considerations, practical challenges, and future research directions aimed at improving efficiency, ensuring interoperability, and expanding lattice cryptography’s use in advanced security frameworks. Through comprehensive analysis, this research underscores the importance of lattice-based cryptography as a critical pillar for securing the digital ecosystem against the emerging threat of quantum computing.
The impending arrival of cryptographically relevant quantum computing threatens classical public‑key infrastructures. This paper reviews the latest developments (2025–2026) in post‑quantum cryptography (PQC), fully homomorphic encryption (FHE), and zero‑knowledge proofs (ZKP). NIST has advanced nine signature candidates to its third evaluation round and selected HQC as a backup encryption standard. Novel primitives include bio‑inspired RNA‑based cryptography, algebraic hash signatures, and topology‑mined lattice schemes. FHE has reached its fifth generation with the GL scheme and the MadPanthera virtual processor, while lightweight ZKPs such as Microsoft’s Vega enable mobile‑friendly verification. These advances demonstrate rapid maturation toward deployable quantum‑safe systems.
Mostly, existing public-key cryptosystems are constructed on the mathematical complexity of large integer factorization and solving discrete logarithms, which will be irresistible to attacks of future quantum computers. Due to the rapidly growing vulnerabilities in cryptographic systems recently, post-quantum cryptography (PQC) has turned out to be a need for the crypto community and has become a necessary research domain for the welfare of society. Lattice-based cryptography is measured as the most capable PQC because of its intense security proofs based on worst-case hardness, balanced performance in security, and efficient computation. This chapter provides a mathematical foundation of lattice-based cryptography, existing digital signatures, different attacks on lattice-based signatures with toy examples, challenges in lattice-based digital signatures, and future research directions. For scholars and academicians functioning in this emerging field of post PQC, the main unresolved issues evaluated in this study will provide an easily navigable reference and a unified, comprehensive source of data on lattice-based cryptography.