In this thesis we will discuss hard computational problems in lattice theory and relate them to cryptographic constructions. Many of these problems enjoy average-case hardness which makes them attractive for cryptography. In addition, lattice based cryptography is a candidate for post-quantum cryptography, as there is no known quantum algorithm which breaks various hardness theorems.\n\nWe build a foundation in algebraic number theory to have the required background to discuss schemes based on discrete algebraic structures. These structures are free Z-modules which permits unique factorization in prime ideals. We relate this algebraic number theory to lattices in R^n so we can use the theory from algebra to our advantage.\n\nWe then define some standard hard computational lattice problems and show how many of these are related to each other. We prove that these problems are at least as hard as finding the shortest vector of a lattice, which we conjecture is computationally infeasible. We then prove a quantum reduction the learning with errors problem, a problem in machine learning . We also show that there is a similar reduction for a variant of this problem over more general rings.
In this thesis we will discuss hard computational problems in lattice theory and relate them to cryptographic constructions. Many of these problems enjoy average-case hardness which makes them attractive for cryptography. In addition, lattice based cryptography is a candidate for post-quantum cryptography, as there is no known quantum algorithm which breaks various hardness theorems.\n\nWe build a foundation in algebraic number theory to have the required background to discuss schemes based on discrete algebraic structures. These structures are free Z-modules which permits unique factorization in prime ideals. We relate this algebraic number theory to lattices in R^n so we can use the theory from algebra to our advantage.\n\nWe then define some standard hard computational lattice problems and show how many of these are related to each other. We prove that these problems are at least as hard as finding the shortest vector of a lattice, which we conjecture is computationally infeasible. We then prove a quantum reduction the learning with errors problem, a problem in machine learning . We also show that there is a similar reduction for a variant of this problem over more general rings.
The security of cryptographic protocols is based on the presumed difficulty of algorithmic problems. Among those identified so far, some of the best problems to serve as a foundation for quantum-proof cryptography come from lattices. Lattices are a mathematical structure defined as a set of space vectors generated by integer combinations of a finite number of linearly independent real vectors (its basis). A typical example of a related security problem is the Shortest Vector Problem (SVP). Given a base of an n-dimensional lattice, find a non-zero short vector. For efficiency reasons, these problems are restricted to lattices arising from number theory, known as structured lattices. As the security assumptions for these particular lattices are different from those for unstructured lattices, it is necessary to study them specifically, which is the aim of this thesis.We have studied the case of NTRU and uSVP modules in rank 2, proving that the SVP problem is equivalent on these two families of lattices. We also show a worst-case to average-case reduction for rank-2 uSVP lattices. Then we show that solving SVP on a prime ideal of small norm is no easier than solving SVP on any ideal.
Lattice-based cryptography is a branch of cryptography exploiting the presumed hardness of some well-known problems on lattices. Its main advantages are its simplicity, efficiency, and apparent security against quantum computers. The principle of the security proofs in lattice-based cryptography is to show that attacking a given scheme is at least as hard as solving a particular problem, as the Learning with Errors problem (LWE) or the Small Integer Solution problem (SIS). Then, by showing that those two problems are at least as hard to solve than a hard problem on lattices, presumed polynomial time intractable, we conclude that the constructed scheme is secure.In this thesis, we improve the foundation of the security proofs and build new cryptographic schemes. We study the hardness of the SIS and LWE problems, and of some of their variants on integer rings of cyclotomic fields and on modules on those rings. We show that there is a classical hardness proof for the LWE problem (Regev's prior reduction was quantum), and that the module variants of SIS and LWE are also hard to solve. We also give two new lattice-based group signature schemes, with security based on SIS and LWE. One is the first lattice-based group signature with logarithmic signature size in the number of users. And the other construction allows another functionality, verifier-local revocation. Finally, we improve the size of some parameters in the work on cryptographic multilinear maps of Garg, Gentry and Halevi in 2013.
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.
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.
Lattice-based cryptography is the use of conjectured hard problems on point lattices in ℝn as the foundation for secure cryptographic systems. Attractive features of lattice cryptography include apparent resistance to quantum attacks (in contrast with most number-theoretic cryptography), high asymptotic efficiency and parallelism, security under worst-case intractability assumptions, and solutions to long-standing open problems in cryptography. This work surveys most of the major developments in lattice cryptography over the past ten years. The main focus is on the foundational short integer solution (SIS) and learning with errors (LWE) problems (and their more efficient ring-based variants), their provable hardness assuming the worst-case intractability of standard lattice problems, and their many cryptographic applications.
Lattice-based cryptography is centered around the hardness of problems on lattices. A lattice is a grid of points that stretches to infinity. With the development of quantum computers, existing cryptographic schemes are at risk because the underlying mathematical problems can, in theory, be easily solved by quantum computers. Since lattice-based mathematical problems are hard to be solved even by quantum computers, lattice-based cryptography is a promising foundation for future cryptographic schemes. In this paper, we focus on lattice-based public-key encryption schemes. This survey presents the current status of the lattice-based public-key encryption schemes and discusses the existing implementations. Our main focus is the learning with errors problem (LWE problem) and its implementations. In this paper, the plain lattice implementations and variants with special algebraic structures such as ring-based variants are discussed. Additionally, we describe a class of lattice-based functions called lattice trapdoors and their applications.