Lattice-Based Cryptosystems and Quantum CryptanalysisQuantum computers are probably coming, though we don't know when-and when they arrive, they will, most likely, be able to break our standard public-key cryptography algorithms.In anticipation of this possibility, cryptographers have been working on quantum-resistant public-key algorithms.The National Institute for Standards and Technology (NIST) has been hosting a competition a since 2017, and there already are several proposed standards.b Most of these are based on lattice problems.The mathematics of lattice cryptography revolve around combining sets of vectors-that's the lattice-in a multi-dimensional space.These lattices are filled with multi-dimensional periodicities.The hard problem c that's used in cryptography is to find the shortest periodicity in a large, random-looking lattice.This can be turned into d a public-key cryptosystem in a variety of different ways.Research has been ongoing since 1996, and there has been some really great work since then-including many practical public-key algorithms.On April 10, Yilei Chen from Tsinghua University in Beijing posted a paper describing a new quantum attack on that shortest-path lattice problem.It's a very dense mathematical paper-63 pages long-and my guess is that only a few cryptographers are able to understand all of its details.(I was not one of them.)But the conclusion was pretty devastating, breaking essentially all of the lattice-based fully homomorphic encryption schemes and coming significantly closer to attacks against the recently proposed (and NIST-approved) lattice key-exchange and signature schemes.However, there was a small but critical mistake in the paper, on the bottom of page 37.It was independently discovered by Hongxun Wu from Berkeley and Thomas Vidick from the Weizmann Institute in Israel eight days later.The attack algorithm in its current form doesn't work.This was discussed last week at the Cryptographers' Panel e at the RSA Conference.Adi Shamir, the "S" in RSA and a 2002 recipient of ACM's A.M. Turing award, described the result as psychologically significant because it shows that there is still a lot to be discovered about quantum
Lattice-Based Cryptosystems and Quantum CryptanalysisQuantum computers are probably coming, though we don't know when-and when they arrive, they will, most likely, be able to break our standard public-key cryptography algorithms.In anticipation of this possibility, cryptographers have been working on quantum-resistant public-key algorithms.The National Institute for Standards and Technology (NIST) has been hosting a competition a since 2017, and there already are several proposed standards.b Most of these are based on lattice problems.The mathematics of lattice cryptography revolve around combining sets of vectors-that's the lattice-in a multi-dimensional space.These lattices are filled with multi-dimensional periodicities.The hard problem c that's used in cryptography is to find the shortest periodicity in a large, random-looking lattice.This can be turned into d a public-key cryptosystem in a variety of different ways.Research has been ongoing since 1996, and there has been some really great work since then-including many practical public-key algorithms.On April 10, Yilei Chen from Tsinghua University in Beijing posted a paper describing a new quantum attack on that shortest-path lattice problem.It's a very dense mathematical paper-63 pages long-and my guess is that only a few cryptographers are able to understand all of its details.(I was not one of them.)But the conclusion was pretty devastating, breaking essentially all of the lattice-based fully homomorphic encryption schemes and coming significantly closer to attacks against the recently proposed (and NIST-approved) lattice key-exchange and signature schemes.However, there was a small but critical mistake in the paper, on the bottom of page 37.It was independently discovered by Hongxun Wu from Berkeley and Thomas Vidick from the Weizmann Institute in Israel eight days later.The attack algorithm in its current form doesn't work.This was discussed last week at the Cryptographers' Panel e at the RSA Conference.Adi Shamir, the "S" in RSA and a 2002 recipient of ACM's A.M. Turing award, described the result as psychologically significant because it shows that there is still a lot to be discovered about quantum
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 invention of public-key cryptography (PKC) by Diffie and Hellman in 1976 is one of two milestones marking the beginning of modern cryptography. The security of a PKC system requires that it should be infeasible to compute the private key from a given public key, which in turn is typically guaranteed by the difficulty in solving some cryptographic-friendly mathematical problems. Among those problems, the integer factorization and discrete logarithm problems play pivotal roles in the development of public-key cryptography. In particular, the assumption that there is no polynomial time (classical) algorithm that solves the above two problems constitutes the basis for the security of almost all currently used public-key cryptosystems, such as RSA and ElGamal. However, Shor [1] found an efficient quantum solving algorithm for the integer factorization and discrete logarithm problems in 1994, which would destroy the security basis for most real deployed PKC systems if large-scale quantum computers become available. The rapid development of quantum technology in recent years suggests that we are getting closer to the quantum crisis of current PKC systems. As a response to the quantum crisis, the research community has made many efforts to post-quantum cryptography (PQC) that are believed to resist quantum computer attacks. Lattice-based cryptography is one of the main directions in this area, and has become the most promising PQC candidate for standardization. The research of lattice-based cryptography dates back to the seminal work of Ajtai [2], which first based the security of cryptographic primitives on the difficulty of solving some lattice problems such as the approximate shortest vector problem and the closest vector problem. After more than two decades of development, lattice-based cryptography has gained substantial progress. In theory, we have witnessed the construction of many powerful cryptographic primitives that were not known before. In practice, the public key size of lattice-based cryptosystems has been significantly reduced from several gigabytes to a few kilobytes. For current applications and standardization, key exchanges, public-key encryptions (PKEs) and signatures are the most desired lattice-based cryptosystems, where the first two are often used to ensure information secrecy, while the last provides information authentication. Note that public-key encryptions are equivalent to two-round key exchanges, also known as the key encapsulation mechanism (KEM). In the following, we focus on lattice-based PKEs/KEMs. The design principle of lattice-based PKEs/KEMs follows two approaches: a trapdoor one-way function and an approximate commutative one-way function. In the first approach, a trapdoor one-way function f and its trapdoor f−1 are generated as the public key and private key. A plaintext m is encrypted as c = f(m), and the ciphertext is decrypted as m = f−1(c). The NTRU scheme and its variants follow such an approach. In the second approach, an approximate commutative function fs and a random input a are generated, the public key is (a, b = fs(a)) and the private key is s. A plaintext m is encrypted as c1 = gr(a), c2 = gr(b) + E(m), where r is a random element, g is an approximate commutative one-way function and E is the encoding of an error-correction code. The ciphertext (c1, c2) is decrypted as m = D(c2 − fs(c1)), where D is the decoding of an error-correction code. The correctness of decryption is guaranteed by the approximate commutative property (gr(b) = gr(fs(a)) ≈ fs(gr(a)) = fs(c1)) and the error-correction code. Learning with errors (LWE) problem-based PKE/KEM schemes and their variants follow such an approach. The standardization of PQC has received substantial support from national funding agencies, such as the European Union projects PQCrypto and SAFEcrypto. The USA National Institute of Standard and Technology (NIST) started the post-quantum standardization project in 2012, organized a worldwide ‘competition’ in 2016 and plans to deliver draft standards before 2024. After two rounds of evaluation, seven schemes were accepted as the round-three candidates on 22 July 2020, of which five are lattice-based schemes. In 2017, the International organization for standardization started a study project of post-quantum cryptography named SD8. Most LWE-based PKE schemes such as Kyber and LAC follow the Lindner-Peikert framework [3] up to the choices of ring structure, noise distribution and parameters. As the size of noise in lattice-based PKEs/EKMs is typically larger than the lower bits of the ciphertext, many schemes often artificially drop a few lower bits to optimize the practical performance without significantly increasing the final decryption noise. Kyber [4] is one of the most promising PKE/KEM candidates in the NIST PQC standardization, whose security is provably based on the difficulty of so-called module LWE (MLWE) problems. The use of MLWE allowed Kyber to obtain additional flexibility and security advantages, while keeping the same efficiency as those based on the ring-LWE problems. The strategy of dropping some lower bits is used to compress both the public-key and ciphertext sizes. For a targeted 128-bit quantum security, the public-key and ciphertext sizes of Kyber is around a single kilobyte. LAC [5] uses several techniques to reduce the modulus as small as possible: it uses a byte-level modulus, embeds powerful error-correcting codes to maximize the noise-modulus rate, provides a different style of modulus to avoid potential attacks for special polynomial ring structures and optimizes the polynomial multiplication algorithm of the bit-level secret and noise. Based on these techniques, LAC reduces the public key to 544–1056 bytes and the ciphertext to 664–1464 bytes. Meanwhile, it achieves similar computational performance compared with NTT technique schemes. By observing the asymmetries in lattice-based cryptosystems, Zhang and Yu et al. [6] proposed asymmetric MLWE and asymmetric MSIS problems, which yield further size-optimized KEMs and signature schemes to those from standard counterparts. Compared to the seven NIST round-three candidates at Category III security, their KEM scheme has parameters (public key + ciphertext size) smaller than Classic McEliece, CRYSTALS-Kyber, NTRU and Saber; their signature scheme has parameters (public key + signature size) smaller than CRYSTALS-Dilithium and Rainbow, but slightly larger than FALCON, where CRYSTALS-Kyber, NTRU, Saber, CRYSTALS-Dilithium and FALCON are lattice-based schemes. One important application of lattice-based PKEs/KEMs is to build authenticated key exchange (AKE), which could provide additional authentication to the established session key. There are several generic ways to construct AKE from lattice-based PKEs/KEMs. In the PKE setting, Xue and Lu et al. [7] proposed a more compact KEM-based framework via a new primitive named ‘2-key KEM’, which shares some item between different ciphertexts. In the only password setting, Zhang and Yu [8] presented a two-round PAKE framework from the splittable-PKEs that can be instantiated from lattices. The only known AKE that does not rely on lattice-based PKEs/KEMs was given in Ref. [9], which has two-round messages and provides implicit authentication. This work was supported by the National Natural Science Foundation of China (6197239 to X.L.; 62022018 and 61932019 to J.Z.) and the National Key Research and Development Program of China (2018YFB0804105 to J.Z.). Conflict of interest statement. None declared.
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.
The standard public-key cryptosystems used today relies mathematical problems that require a lot of computing force to solve, so much that, with the right parameters, they are computationally unsolvable. But there are quantum algorithms that are able to solve these problems in much shorter time. These quantum algorithms have been known for many years, but have only been a problem in theory because of the lack of quantum computers. But with recent development in the building of quantum computers, the cryptographic world is looking for quantum-resistant replacements for today’s standard public-key cryptosystems. Public-key cryptosystems based on lattices are possible replacements. This thesis presents several possible candidates for new standard public-key cryptosystems, mainly NTRU and ring-LWE-based systems. The latticebased cryptosystems are shown to be very fast and have strong, provable security against quantum computers, but are a lot more complicated than RSA and Diffie-Hellman.
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.
In 1994, P. Shor discovered quantum algorithms that can break both the RSA cryptosystem and the ElGamal cryptosystem. In 2007, D-Wave demonstrated the first quantum computer. These events and further developments have brought a crisis to secret communication. In 2016, the National Institute of Standards and Technology (NIST) launched a global project to solicit and select a handful of encryption algorithms with the ability to resist quantum computer attacks. In 2022, it announced four candidates, CRYSTALS-Kyber, CRYSTALS-Dilithium, Falcon, and Sphincs+, for post-quantum cryptography standards. The first three are based on lattice theory and the last on a hash function. The security of lattice-based cryptosystems relies on the computational complexity of the shortest vector problem (SVP), the closest vector problem (CVP), and their generalizations. As we will explain, the SVP is a ball-packing problem, and the CVP is a ball-covering problem. Furthermore, both the SVP and CVP are equivalent to arithmetic problems for positive definite quadratic forms. This paper will briefly describe the mathematical problems on which lattice-based cryptography is built so that cryptographers can extend their views and learn something useful.