This chapter introduces post-quantum cryptography. It describes lattice-based cryptographic constructions of post-quantum cryptography. Lattice-based cryptographic constructions are one of candidates of post-quantum cryptography, which are believed to be secure against quantum computers. Learning with errors (LWE) is one of the most promising primitives in many usages due to its lightweight operation and rigorous security reduction against the worst-case of the lattice problems that are considered to be hard to solve even after the advance of quantum computers. After the research about the connection between the LWE problem and some lattice problems, some variants of LWE, of which the secret distributions are modified from the uniform distribution, were proposed. Due to the efficiency and compactness of ring-LWE, many lattice-based cryptosystems are constructed as ring-LWE based, rather than LWE-based. The chapter explains a ring variant of Lizard scheme, called RLizard, IND-CCA secure encryption scheme.
This chapter introduces post-quantum cryptography. It describes lattice-based cryptographic constructions of post-quantum cryptography. Lattice-based cryptographic constructions are one of candidates of post-quantum cryptography, which are believed to be secure against quantum computers. Learning with errors (LWE) is one of the most promising primitives in many usages due to its lightweight operation and rigorous security reduction against the worst-case of the lattice problems that are considered to be hard to solve even after the advance of quantum computers. After the research about the connection between the LWE problem and some lattice problems, some variants of LWE, of which the secret distributions are modified from the uniform distribution, were proposed. Due to the efficiency and compactness of ring-LWE, many lattice-based cryptosystems are constructed as ring-LWE based, rather than LWE-based. The chapter explains a ring variant of Lizard scheme, called RLizard, IND-CCA secure encryption scheme.
The swift advancement of quantum computing devices holds the potential to create robust machines that can tackle an extensive array of issues beyond the scope of conventional computers. Consequently, quantum computing machines create new risks at a velocity and scale never seen before, especially with regard to encryption. Lattice-based cryptography is regarded as post-quantum cryptography’s future and a competitor to a quantum computer attack. Thus, there are several advantages to lattice-based cryptographic protocols, including security, effectiveness, reduced energy usage and speed. In this work, we study the learning with errors (LWE) problem and the cryptosystems that are based on the LWE problem and, in addition, we present a new efficient variant of LWE cryptographic scheme.
Lattice-based cryptography seems to be a good area for the researchers in cryptology because of its vulnerability against quantum computing attacks. Such resources as extensive documentation, theoretical foundations, and practical implementations are provided with respect to some key algorithms, which are Learning With Errors (LWE) and Ring-LWE, and sophisticated ones such as homomorphic encryption. This paper aims to make academic research more practical by providing tools for cryptographers and programmers to experiment with lattice-based cryptography in secure communications and data protection. Thus, we examine and implement Lizard, Ring-Lizard, Kyber, Frodo, and NewHope in JavaScript, which are current postquantum cryptography possibilities due to their applicability and efficiency. In this case quantum computational capabilities are referred to as those that may compromise security thus undermining post-quantum cryptography.
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.
Post-quantum cryptography (PQC) aims to develop cryptographic protocols resistant to attacks by quantum computers, which threaten classical schemes based on integer factorization and discrete logarithm problems. Among various approaches, lattice-based cryptography has emerged as one of the most promising candidates for PQC. Within this domain, ring theory plays a foundational role by providing the algebraic structures—specifically polynomial rings modulo cyclotomic polynomials—used to define hard problems like Ring Learning With Errors (Ring-LWE). These problems underpin efficient and secure cryptographic primitives such as key exchange, encryption, digital signatures, and homomorphic encryption. This paper explores the theoretical background of ring theory relevant to PQC and discusses how it enables the construction of cryptosystems with strong security assumptions and practical efficiency. We also analyze specific schemes that utilize ring-based lattices, including New Hope, NTRUEncrypt, and Dilithium, highlighting the advantages of ring structures in reducing key sizes and improving computational speed. Furthermore, we review the implementation challenges and potential future directions for integrating ring theory into next-generation cryptographic protocols suitable for a post-quantum world.
The advent of quantum computing poses a significant existential threat to current public-key cryptographic systems, most notably RSA and Elliptic Curve Cryptography (ECC), which rely on the presumed hardness of integer factorization and discrete logarithm problems. Shor's algorithm, a quantum algorithm, can efficiently solve these problems, rendering classical asymmetric primitives insecure. This paper addresses the imperative need for post-quantum cryptography (PQC) by exploring lattice-based constructs as robust alternatives for asymmetric primitives. We provide a comprehensive overview of the fundamental mathematical problems underlying lattice-based cryptography, such as the Shortest Vector Problem (SVP), Closest Vector Problem (CVP), Learning With Errors (LWE), and Ring Learning With Errors (RLWE). The paper surveys prominent lattice-based schemes, including NTRU, CRYSTALS-Kyber, and CRYSTALS-Dilithium, analyzing their security foundations, performance characteristics, and the extent to which they offer a viable "post-quantum RSA." We delve into the methodology of constructing these primitives, discuss their resilience against both classical and quantum attacks, and evaluate their practical implications for key sizes, computational efficiency, and deployment challenges. Our analysis highlights that lattice-based cryptography, particularly the NIST-standardized schemes, presents a strong candidate for securing digital communications in the quantum era, offering provable security rooted in worst-case to average-case reductions for lattice problems.
Among the different cryptosystems, the Public Key Cryptosystems (PKCS) are of great usage and are used in many applications. Most of the current PKCS are based on a hard problem making the cryptanalysis of such cipher infeasible given the current computing and memory availability. However, Quantum Computing threatens all the current PKCS as they offer a new model that can solve those hard problems, e.g. the factorization problem of RSA or the discrete logarithm problem of ELGAMAL algorithm. Lattice Based PKCS are a promising filed to introduce an immune cipher system against quantum cryptanalysis. This paper presents a summary of recent progress in the field of lattice-based cryptography, by focusing on the recent advancement in lattice-based public key cryptosystems and key exchange mechanisms based on the learning with errors (LWE) problem and its ring variant Ring-LWE.