Digital signatures are an important primitive for building secure systems and are used in most real-world security protocols. However, almost all popular signature schemes are either based on the factoring assumption (RSA) or the hardness of the discrete logarithm problem (DSA/ECDSA). In the case of classical cryptanalytic advances or progress on the development of quantum computers, the hardness of these closely related problems might be seriously weakened. A potential alternative approach is the construction of signature schemes based on the hardness of certain lattice problems that are assumed to be intractable by quantum computers. Due to significant research advancements in recent years, lattice-based schemes have now become practical and appear to be a very viable alternative to number-theoretic cryptography. In this article, we focus on recent developments and the current state of the art in lattice-based digital signatures and provide a comprehensive survey discussing signature schemes with respect to practicality. Additionally, we discuss future research areas that are essential for the continued development of lattice-based cryptography.
Digital signatures are an important primitive for building secure systems and are used in most real-world security protocols. However, almost all popular signature schemes are either based on the factoring assumption (RSA) or the hardness of the discrete logarithm problem (DSA/ECDSA). In the case of classical cryptanalytic advances or progress on the development of quantum computers, the hardness of these closely related problems might be seriously weakened. A potential alternative approach is the construction of signature schemes based on the hardness of certain lattice problems that are assumed to be intractable by quantum computers. Due to significant research advancements in recent years, lattice-based schemes have now become practical and appear to be a very viable alternative to number-theoretic cryptography. In this article, we focus on recent developments and the current state of the art in lattice-based digital signatures and provide a comprehensive survey discussing signature schemes with respect to practicality. Additionally, we discuss future research areas that are essential for the continued development of lattice-based cryptography.
Quantum computers are moving closer to reality, and with that comes a serious concern for today’s public-key cryptography. Many of the encryption and digital signature systems we rely on now, such as RSA and ECC, may no longer be secure in the future. Because of this, researchers are actively looking for new methods that can protect digital communication in the post-quantum era. One of the most promising directions is lattice-based cryptography. This paper gives a simple and focused review of lattice-based signature schemes. It brings together the main ideas from the existing literature and explains how these schemes are built, why they are considered secure, and what makes them practical. The review mainly discusses schemes based on hard lattice problems such as SIS and LWE. To make the comparison clearer, a table is also included that compares different signature schemes in terms of key size, signature size, security level, and computational efficiency. Overall, this study shows why lattice-based signature schemes are becoming an important part of modern cryptographic research and why they may play a major role in securing digital systems against future quantum threats.
Quantum computers are moving closer to reality, and with that comes a serious concern for today's public-key cryptography. Many of the encryption and digital signature systems we rely on now, such as RSA and ECC, may no longer be secure in the future. Because of this, researchers are actively looking for new methods that can protect digital communication in the post-quantum era. One of the most promising directions is lattice-based cryptography. This paper gives a simple and focused review of lattice-based signature schemes. It brings together the main ideas from the existing literature and explains how these schemes are built, why they are considered secure, and what makes them practical. The review mainly discusses schemes based on hard lattice problems such as SIS and LWE. To make the comparison clearer, a table is also included that compares different signature schemes in terms of key size, signature size, security level, and computational efficiency. Overall, this study shows why lattice-based signature schemes are becoming an important part of modern cryptographic research and why they may play a major role in securing digital systems against future quantum threats.
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.
Abstract Lattice-based digital signature has become one of the widely recognized post-quantum algorithms because of its simple algebraic operation, rich mathematical foundation and worst-case security, and also an important tool for constructing cryptography. This survey explores lattice-based digital signatures, a promising post-quantum resistant alternative to traditional schemes relying on factoring or discrete logarithm problems, which face increasing risks from quantum computing. The study covers conventional paradigms like Hash-and-Sign and Fiat-Shamir, as well as specialized applications including group, ring, blind, and proxy signatures. It analyzes the versatility and security strengths of lattice-based schemes, providing practical insights. Each chapter summarizes advancements in schemes, identifying emerging trends. We also pinpoint future directions to deploy lattice-based digital signatures including quantum cryptography.
Digital signatures are indispensable for security on the Internet, because they guarantee authenticity, integrity, and non-repudiation, of namely e-mails, software updates, and in the Transport Layer Security (TLS) protocol which is used for secure data transfer, for example. Most signature schemes that are currently in use such as the RSA signature scheme, are considered secure as long as the integer factorization problem or the discrete logarithm (DL) problem are computationally hard. At present, no algorithms have yet been found to solve these problems on conventional computers in polynomial time. However, in 1997, Shor published a polynomial-time algorithm that uses quantum computation to solve the integer factorization and the DL problem. In particular, this means that RSA signatures are considered broken as soon as large-scale quantum computers exist. Due to significant advances in the area of quantum computing, it is reasonable to assume that within 20 years, quantum computers that are able to break the RSA scheme, could exist. In order to maintain authenticity, integrity, and non-repudiation of data, cryptographic schemes that cannot be broken by quantum attacks are required. In addition, these so-called post-quantum secure schemes should be sufficiently efficient to be suitable for all established applications. Furthermore, solutions enabling a timely and secure transition from classical to post-quantum schemes are needed. This thesis contributes to the above-mentioned transition. In this thesis, we present the two lattice-based digital signature schemes TESLA and qTESLA, whereby lattice-based cryptography is one of five approaches to construct post-quantum secure schemes. Furthermore, we prove that our signature schemes are secure as long as the so-called Learning With Errors (LWE) problem is computationally hard to solve. It is presumed that even quantum computers cannot solve the LWE problem in polynomial time. The security of our schemes is proven using security reductions. Since our reductions are tight and explicit, efficient instantiations are possible that provably guarantee a selected security level, as long as the corresponding LWE instance provides a certain hardness level. Since both our reductions (as proven in the quantum random oracle model) and instantiations, take into account quantum attackers, TESLA and qTESLA are considered post-quantum secure. Concurrently, the run-times for generating and verifying signatures of qTESLA are similar (or faster) than those of the RSA scheme. However, key and signature sizes of RSA are smaller than those of qTESLA. In order to protect both the theoretical signature schemes and their implementations against attacks, we analyze possible vulnerabilities against implementation attacks. In particular, cache-side-channel attacks resulting from observing the cache behavior and fault attacks, which recover secret information by actively disrupting the execution of an algorithm are focused. We present effective countermeasures for each implementation attack we found. Our analyses and countermeasures also influence the design and implementation of qTESLA. Although our schemes are considered (post-quantum) secure according to state-of-the-art LWE attacks, cryptanalysis of lattice-based schemes is still a relatively new field of research in comparison to RSA schemes. Hence, there is a lack of confidence in the concrete instantiations and their promised security levels. However, due to developments within the field of quantum computers, a transition to post-quantum secure solutions seems to be more urgently required than ever. To solve this dilemma, we present an approach to combine two schemes, e.g., qTESLA and the RSA signature scheme, so that the combination is secure as long as one of the two combined schemes is secure. We present several of such combiners to construct hybrid signature schemes and hybrid key encapsulation mechanisms to ensure both authenticity and confidentiality in our Public-Key Infrastructure (PKI). Lastly, we also demonstrate how to apply the resulting hybrid schemes in standards such as X.509 or TLS. To summarize, this work presents post-quantum secure candidates which can, using our hybrid schemes, add post-quantum security to the current classical security in our PKI.
Building cryptographic schemes upon as many fundamentally different hard problems as possible, seems to be the best way to hedge against future threats such as quantum computers. Being mainly based on the hardness of factoring and computing discrete logarithms, the present security landscape is at risk. In contrast, problems in lattices, such as finding short non-zero vectors, seem to withstand quantum computer attacks and the best known algorithms run in exponential time. In sharp contrast to other fields of cryptography, lattices admit a worst-case to average-case reduction (Ajtai 1996). Instead of assuming that a problem is hard for randomly chosen instances, lattice-based cryptosystems merely require the existence of a single hard instance, i.e., hardness in the worst case. With such an additional "trust anchor", the resulting security guarantees are much more plausible. Quite recently, we have seen an increased interest in lattice-based cryptography with many striking results. In this thesis, we are particularly interested in signature schemes, which provide a supporting pillar for today's economy. While we have seen basic signature schemes from lattices, e.g., (Gentry, Peikert, Vaikuntanathan 2008), (Lyubashevsky, Micciancio 2008), (Lyubashevsky 2009), or (Cash, Hofheinz, Kiltz, Peikert 2009), there are hardly any results dealing with the specific needs of applications, where ordinary signatures often fall too short. In this thesis, we build upon the above results and equip them with additional features, motivated by an exemplary selection of application scenarios. Hence, we demonstrate the great versatility of lattices in cryptography. In particular, we facilitate privacy-friendly electronic elections, fair online contract signing, signature compression, secure signatures in the strongest sense, as well as identity-based primitives. As far as possible, we avoid simplifying assumptions, such as the random oracle model. We believe that our techniques can be transferred to other application scenarios as well. Independently of the these results, we discuss the practical hardness of lattice problems and provide a framework for estimating the security of essentially all modern lattice-based cryptography.