The threat of large-scale quantum computers motivates cryptographers to base cryptosystems on problems believed to be resistant against quantum computers. In this thesis, we focus on the LWE problem which is believed to be resistant against quantum computers. First, we describe lattices which are closely related to the LWE problem. We introduce basic notions, describe lattice problems and solve exercises related to the covering radius of lattice. After that, we introduce the LWE problem and its variants. We prove reductions from two lattice problems to certain variant of the LWE problem. We define the notion of statistical distance and prove some lemmata about it which we need within reductions. Moreover, we show concrete application of the LWE problem. We describe a scheme for key exchange and briefly prove its security under the assumption that the LWE problem is hard. 1
The threat of large-scale quantum computers motivates cryptographers to base cryptosystems on problems believed to be resistant against quantum computers. In this thesis, we focus on the LWE problem which is believed to be resistant against quantum computers. First, we describe lattices which are closely related to the LWE problem. We introduce basic notions, describe lattice problems and solve exercises related to the covering radius of lattice. After that, we introduce the LWE problem and its variants. We prove reductions from two lattice problems to certain variant of the LWE problem. We define the notion of statistical distance and prove some lemmata about it which we need within reductions. Moreover, we show concrete application of the LWE problem. We describe a scheme for key exchange and briefly prove its security under the assumption that the LWE problem is hard. 1
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 only known threat to some symmetric cryptographic schemes like DES is Grover’s algorithm that proposes a square root speed-up over classical brute force algorithms. NIST [14] points out that if the key sizes are sufficient, symmetric cryptographic schemes (specifically the Advanced Encryption Standard-AES) are resistant to quantum computers. The development of quantum computer is on full pace, hopefully it might be available in a decade. Thus, it is necessary to develop efficient quantum secure public key cryptosystems to provide safe key exchange in quantum era. In this paper we presented an efficient implementation of a Ring-Learning with errors public key cryptosystem whose security is based on intractability of hard problem on lattices. And comparative analysis of proposed cryptosystem with RSA, ECDH and LWE based key exchange protocols is presented.
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.
In 1996 Hoffstein, Pipher ad Silverman presented NTRUEncrypt, which is to
date the fastest known lattice-based encryption scheme. Its moderate
key-sizes, excellent asymptotic performance and conjectured resistance to
quantum attacks make it a perfect candidate to succeed where factorization and discrete log fail. Unfortunately, no security proof has been produced for NTRUEncrypt nor for its signature counterpart NTRUSign.
In 2013 Stehle and Steinfield proposed to apply some mild modification to
the encryption and signature scheme to make them provably secure, under the assumed quantum hardness of standard worst-case lattice problems, restricted to a family of lattices related to some cyclotomic fields. In particular they showed that if the secret key polynomials of the encryption scheme are chosen from discrete Gaussians, then the public key, i.e their ratio, is statistically indistinguishable from uniform. The security will then follow from the hardness of the R-LWE problem.The aim of this thesis is to present Stehle's and Steinfield's work in a slightly more accessi-ble form, providing some more background and details in some points.
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.