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.
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.
The landscape of modern cryptography is fundamentally challenged by the advent of quantum computing. Current public-key cryptosystems, which form the bedrock of secure digital communication, rely on the computational hardness of problems such as integer factorization (RSA) and the discrete logarithm problem (ECC). However, Shor's algorithm, if implemented on a sufficiently powerful quantum computer, can efficiently break these schemes, rendering them insecure. This paper explores the critical transition from these classical number theory problems to lattice-based cryptography, presenting a comprehensive analysis of the motivations, theoretical underpinnings, and practical implications for forging post-quantum cryptographic security. We delve into the mathematical foundations of lattice problems, such as the Shortest Vector Problem (SVP) and Learning With Errors (LWE), which are believed to resist known quantum algorithms. Through a thorough literature review, we trace the evolution of cryptographic thought, detailing the vulnerabilities of pre-quantum schemes and the emergence of lattice-based constructions as leading candidates for post-quantum standardization. The methodology outlines a comparative theoretical analysis of security, efficiency, and robustness against both classical and quantum attacks. Results demonstrate the compelling advantages of lattice-based schemes in terms of provable security reductions and resistance to quantum threats, albeit with considerations for increased key and ciphertext sizes. The discussion highlights the challenges and opportunities in the deployment and standardization of these new cryptographic primitives, emphasizing their critical role in securing future digital infrastructures. This work concludes that lattice-based cryptography offers a robust and theoretically sound pathway to establishing enduring security in a quantum-threatened world.
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.
The advent of quantum computing poses a significant existential threat to current public-key cryptographic standards, particularly RSA and Elliptic Curve Cryptography (ECC), which underpin global digital security. Shor's algorithm, if implemented on a sufficiently powerful quantum computer, can efficiently break these foundational schemes, rendering vast amounts of encrypted data vulnerable. This paper investigates lattice-based cryptography as a robust and promising candidate for post-quantum security. We provide a comprehensive analysis of its theoretical underpinnings, focusing on hard problems like the Shortest Vector Problem (SVP) and Learning With Errors (LWE), which are believed to be intractable even for quantum adversaries. Through a detailed literature review and methodological exposition, we examine the construction principles of prominent lattice-based Key Encapsulation Mechanisms (KEMs) and digital signature schemes, such as Kyber and Dilithium, highlighting their inherent security properties and performance characteristics. Our results demonstrate that lattice-based schemes offer competitive efficiency, compact key sizes, and strong security guarantees against both classical and quantum attacks, positioning them as essential building blocks for the post-RSA cryptographic landscape. The discussion elaborates on the practical implications, standardization efforts by NIST, and the challenges associated with the transition to quantum-resilient cryptographic infrastructure. We conclude that lattice-based cryptography is a cornerstone of quantum resilience, critical for securing future digital communications and data integrity.
The advent of large-scale quantum computers poses a profound and imminent threat to the security foundations of modern cryptography. Widely deployed public-key cryptosystems, such as RSA and Elliptic Curve Cryptography (ECC), rely on problems like integer factorization and discrete logarithms, which Shor's algorithm can efficiently solve. This necessitates a proactive transition to post-quantum cryptography (PQC). Among leading candidates, lattice-based cryptography stands out due to its robust security guarantees, based on hard problems lacking known efficient quantum solutions, and structural properties facilitating efficient implementation. This paper explores the fundamental principles of lattice-based cryptography, detailing underlying mathematical hard problems such as the Shortest Vector Problem (SVP) and Learning With Errors (LWE). We review the evolution of lattice-based schemes, from early constructions to contemporary candidates like Kyber and Dilithium, emphasizing their suitability for Key Encapsulation Mechanisms (KEMs) and Digital Signature Schemes (DSS). Furthermore, we analyze their performance characteristics, security assumptions, and resilience against classical and quantum attacks, providing a critical comparison with other PQC families. This comprehensive overview, structured through a systematic critical analysis framework, highlights the ascendancy of lattice-based cryptography as a cornerstone for future quantum-resistant security infrastructure, offering theoretical underpinnings, practical implications, and actionable future research directions for a post-quantum world.
The advent of large-scale quantum computers poses an existential threat to the security of current public-key cryptosystems, such as RSA and Elliptic Curve Cryptography (ECC), which rely on the computational intractability of integer factorization and discrete logarithms. This paper explores lattice-based cryptography as a promising candidate for post-quantum security. We delve into the mathematical foundations of lattices, specifically focusing on hard problems like the Shortest Vector Problem (SVP), Closest Vector Problem (CVP), Small Integer Solution (SIS), and Learning With Errors (LWE) problems, which are believed to remain intractable even for quantum computers. We provide a comprehensive review of prominent lattice-based constructions, including NTRU, Kyber, and Dilithium, highlighting their security proofs, efficiency considerations, and suitability for various cryptographic primitives such as public-key encryption, key-exchange mechanisms, and digital signatures. Furthermore, we discuss the challenges and opportunities associated with their deployment, including parameter selection, resistance to side-channel attacks, and integration into existing cryptographic infrastructures. By examining the theoretical underpinnings and practical implications, this work aims to solidify understanding of lattice-based methods as a robust and essential component of the cryptographic landscape in the quantum era, moving beyond the traditional RSA frontier.
The advent of quantum computing poses a significant threat to the security of current asymmetric cryptographic systems, which underpin much of modern digital communication and security. These systems, predominantly RSA and Elliptic Curve Cryptography (ECC), rely on the computational hardness of mathematical problems such as integer factorization and the discrete logarithm problem. Shor's algorithm, a quantum algorithm, demonstrates the ability to efficiently solve these problems, rendering classical public-key cryptography vulnerable to attacks by sufficiently powerful quantum computers. This paper argues for and thoroughly examines the strategic imperative of transitioning from these classical cryptographic foundations to a new paradigm based on lattices, which are believed to be resistant to quantum attacks. We delve into the fundamental mathematical problems underlying lattice-based cryptography, such as the Shortest Vector Problem (SVP) and the Learning With Errors (LWE) problem, whose worst-case hardness is leveraged to achieve cryptographic security. A comprehensive literature review is provided, tracing the evolution of quantum threats and the emergence of lattice-based schemes. The methodology details the construction principles of prominent lattice-based cryptosystems. Results demonstrate their theoretical security against quantum adversaries and highlight their practical attributes, including compact key sizes and competitive performance. The discussion section contextualizes lattice-based cryptography within the broader landscape of post-quantum candidates, examining their strengths, weaknesses, and the ongoing standardization efforts by organizations like NIST. We conclude that lattice-based cryptography offers a robust and promising foundation for building resilient asymmetric cryptographic systems in the post-quantum era, ensuring the long-term security of digital information through a necessary re-architecture of cryptographic primitives.