This thesis studies the security of lattice-based and code-based cryptography, two leading approaches to post-quantum cryptography. Although several lattice-based and code-based schemes have been selected for standardization and deployment, these schemes and their underlying computational problems have received less scrutiny than those they aim to replace. Continued cryptanalysis is therefore essential to build confidence in the security of post-quantum cryptography. The first part of this thesis concerns the classical cryptanalysis of lattice-based cryptography. We provide an algorithm that solves the Short Integer Solution problem in subexponential time for nontrivial parameter regimes. Moreover, we develop a predictive model for analyzing the practical performance of module-BKZ, a module-lattice analog of the BKZ algorithm. The second part considers the quantum cryptanalysis of lattice-based and code-based cryptography, focusing on quantum speedups for sieving algorithms. We present a new quantum algorithm for 3-tuple lattice sieving that yields an improved time-memory trade-off for solving the Shortest Vector Problem. We also introduce quantum algorithms for code sieving and investigate their potential for attacking code-based cryptography. Together, these results provide new insights into the security of lattice-based and code-based cryptography and highlight several directions for future research.
This thesis studies the security of lattice-based and code-based cryptography, two leading approaches to post-quantum cryptography. Although several lattice-based and code-based schemes have been selected for standardization and deployment, these schemes and their underlying computational problems have received less scrutiny than those they aim to replace. Continued cryptanalysis is therefore essential to build confidence in the security of post-quantum cryptography. The first part of this thesis concerns the classical cryptanalysis of lattice-based cryptography. We provide an algorithm that solves the Short Integer Solution problem in subexponential time for nontrivial parameter regimes. Moreover, we develop a predictive model for analyzing the practical performance of module-BKZ, a module-lattice analog of the BKZ algorithm. The second part considers the quantum cryptanalysis of lattice-based and code-based cryptography, focusing on quantum speedups for sieving algorithms. We present a new quantum algorithm for 3-tuple lattice sieving that yields an improved time-memory trade-off for solving the Shortest Vector Problem. We also introduce quantum algorithms for code sieving and investigate their potential for attacking code-based cryptography. Together, these results provide new insights into the security of lattice-based and code-based cryptography and highlight several directions for future research.
The landscape of modern cryptography, traditionally anchored in the computational hardness of number-theoretic problems such as integer factorization and the discrete logarithm problem, faces an existential threat from the advent of quantum computing. Shor's algorithm, discovered in 1994, demonstrates the theoretical capability of a sufficiently powerful quantum computer to efficiently solve these problems, rendering widely used cryptographic schemes like RSA and Elliptic Curve Cryptography (ECC) insecure. This imminent threat necessitates a paradigm shift towards post-quantum cryptography (PQC), which aims to develop cryptographic algorithms resistant to both classical and quantum attacks. Among the various candidates for PQC, lattice-based cryptography has emerged as a particularly promising area. This paper provides a comprehensive overview of the transition from classical number theory-based cryptography to lattice-based schemes. We explore the fundamental mathematical principles underlying lattices, introduce key lattice problems such as the Shortest Vector Problem (SVP), Closest Vector Problem (CVP), Small Integer Solution (SIS), and Learning With Errors (LWE), and discuss their application in constructing quantum-resistant cryptographic primitives. Furthermore, we analyze the security, efficiency, and practical considerations of prominent lattice-based cryptosystems, highlighting their role in the ongoing NIST post-quantum cryptography standardization process. The paper concludes by discussing the challenges and future directions in the deployment and research of lattice-based cryptographic solutions.
The lattice-based encryption paradigm (LBE) represents a new security model emphasizing flexible development, high levels of security, and efficient computations. It is now becoming more commonly seen as an alternative to other systems of post-quantum cryptography due to its foundations for both security (i.e., there is a high degree of confidence that lattice-based cryptography is secure) and efficiency with respect to computation. The theoretical basis for this report is related to the difficulty of solving the short vector problem (SVP) and other cryptographic hard problems concerning lattices. The report focuses on the hardness of lattice-based cryptography relative to three sub-components of the theoretical work: the relationship between geometric number theory and lattice-based systems, the application of AI to help determine parameters and the efficiencies of finding new lattice reduction algorithms, and finally estimating the difficulty to solve lattice problems. This report reviews the inherent difficulty of short vector problems (SVPs) and other kinds of SVPs, such as the closest vector problem. The report analyzes how to reduce both the time and space complexities for classical and quantum computing solutions through the discovery of new techniques based upon number-theoretic concepts. The report also discusses a number of examples, including digital signature products, encryption systems and homomorphic encryption systems as primitive constructions of cryptography based on lattices or through the properties of the lattice structures. Finally, based on number-theoretical relationships among complex numbers, the report provides a new theoretical approach to the construction of lattice-based systems and their strength relative to other types of cryptographic systems via the provision of theoretical bounds related to the lengths of lattice basis vectors.As a result of this research into bridging the divide between theoretical and practical cryptography, the results show that lattice-based systems have the capability of providing the strength to withstand quantum attacks. Through this research study, there is an improvement in how we understand mathematically the difficulty of lattices and how to create post-quantum secure cryptographic techniques through the use of lattices...
The rapid development of quantum computing technologies poses a serious threat to modern public-key cryptographic systems. Classical cryptographic algorithms such as RSA and elliptic-curve cryptography rely on mathematical problems that can be efficiently solved using quantum algorithms. Post-quantum cryptography (PQC) aims to develop cryptographic schemes that remain secure even in the presence of quantum adversaries. This paper investigates the mathematical foundations of PQC, focusing on lattice-based, code-based, and hash-based cryptographic constructions. Security models based on computational complexity theory are analyzed, and illustrative examples using Learning With Errors (LWE) and decoding problems are presented. Experimental analysis demonstrates that PQC schemes provide strong resistance against both classical and quantum attacks when appropriate parameters are selected. The results highlight the importance of adopting new cryptographic standards to ensure long-term data security in quantum computing environments.
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 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 The paper provides an overview of the main approaches to the construction of post-quantum cryptographic systems that are currently used. The area of lattice-based cryptography is analyzed in detail. We give the description and characterization of some known lattice-based cryptosystems whose resilience is based on the complexity of the shortest vector problem, learning with errors problem, and their variations. The main approaches to solving the problems from lattice theory, on which attacks on the corresponding cryptosystems are based, are analyzed. In particular, some known theoretical estimates of time and memory complexity of lattice basis reduction and lattice sieving algorithms are presented.