The development of quantum-resistant solutions is imperative as the emergence of quantum computing presents a substantial risk to existing cryptography systems. Lattice-based cryptography, especially schemes based on the Learning with Errors (LWE) problem, is one of the most promising methods. To guarantee long-term security, even LWE-based methods could need to be strengthened further as quantum algorithms advance. By mixing Gaussian and discrete uniform distributions to create a mixed error distribution, this work enhances the classic LWE problem. The experimental findings show that, with a slight rise in computing overhead, the mixed error distribution improves the security of the LWE problem by strengthening its resistance to quantum techniques. By presenting a novel approach for enhancing the resilience of cryptographic methods in the quantum era, this research contributes to the continuing work in post-quantum cryptography. Moreover, it introduces the direction of future model improvements and provides multidisciplinary methods for increasing the complexity of cryptographic algorithms.
The development of quantum-resistant solutions is imperative as the emergence of quantum computing presents a substantial risk to existing cryptography systems. Lattice-based cryptography, especially schemes based on the Learning with Errors (LWE) problem, is one of the most promising methods. To guarantee long-term security, even LWE-based methods could need to be strengthened further as quantum algorithms advance. By mixing Gaussian and discrete uniform distributions to create a mixed error distribution, this work enhances the classic LWE problem. The experimental findings show that, with a slight rise in computing overhead, the mixed error distribution improves the security of the LWE problem by strengthening its resistance to quantum techniques. By presenting a novel approach for enhancing the resilience of cryptographic methods in the quantum era, this research contributes to the continuing work in post-quantum cryptography. Moreover, it introduces the direction of future model improvements and provides multidisciplinary methods for increasing the complexity of cryptographic algorithms.
With the fast-paced development of quantum computing, classical cryptographic techniques are under severe security threats, prompting the need for post-quantum cryptographic mechanisms for secure data exchange in next-generation network communications. Post-quantum cryptography seeks to protect sensitive data against quantum attacks to provide long-term data confidentiality. But all current cryptographic techniques, including RSA and ECC, are susceptible to quantum algorithms like Shor's algorithm, which can factor large integers efficiently and break discrete logarithm problems, making classical cryptography useless. There is thus a pressing need for quantum attack-resistant encryption mechanisms. To meet these challenges, we introduce the Lattice-Based Cryptography with Hybrid Key Exchange (L-CHKE) framework, which combines lattice-based encryption with a hybrid key exchange mechanism. Lattice-based cryptography provides quantum resistance because it is based on hard mathematical problems such as Learning with Errors (LWE), which provides greater security and efficiency. The designed L-CHKE algorithm strengthens secure key distribution and authentication for next-generation networks, especially for applications needing strong security like financial transactions, IoT, and military communications. Experimental results verify that L-CHKE noticeably boosts computational efficiency by 98.45%, decreases key exchange latency by 42.56%, and guarantees robustness by 97.64% against quantum attacks, indicating it is an appropriate solution for post-quantum secure communication.
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.
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 advancement of quantum computing poses a significant threat to classical cryptographic algorithms, particularly those based on factorization and discrete logarithm problems. In this paper, we propose a secure and quantum-resistant key exchange scheme leveraging the FrodoKEM algorithm, a lattice-based post-quantum key encapsulation mechanism designed to resist attacks from quantum adversaries. FrodoKEM, being based on the Learning with Errors (LWE) problem, offers a conservative and standard approach to post-quantum cryptography without relying on structured lattices. We compare the performance and security of the proposed scheme against traditional key exchange mechanisms such as RSA and Elliptic Curve Diffie-Hellman (ECDH), as well as other post-quantum algorithms like Kyber and NTRU. Our analysis demonstrates that FrodoKEM offers improved security assurances and competitive performance metrics, especially in environments where long-term confidentiality and forward secrecy are critical. The experimental results affirm FrodoKEM's suitability for real-world deployment in post-quantum secure communications.
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.
Digital signatures and encryption schemes constitute arguably an integral part of cryptographic schemes with the goal to meet the security needs of present and future private and business applications. However, almost all public key cryptosystems applied in practice are put at risk due to its vulnerability to quantum attacks as a result of Shor's quantum algorithm. The magnitude of economic and social impact is tremendous inherently asking for alternatives replacing classical schemes in case large-scale quantum computers are built. Lattice-based cryptography emerged as a powerful candidate attracting lots of attention not only due to its conjectured resistance against quantum attacks, but also because of its unique security guarantee to provide worst-case hardness of average-case instances. Hence, the requirement of imposing further assumptions on the hardness of randomly chosen instances disappears, resulting in more efficient instantiations of cryptographic schemes. The best known lattice attack algorithms run in exponential time. \nIn this thesis we contribute to a smooth transition into a world with practically efficient lattice-based cryptographic schemes. This is indeed accomplished by designing new algorithms and cryptographic schemes as well as improving existing ones. Our contributions are threefold. \nFirst, we construct new encryption schemes that fully exploit the error term in LWE instances. To this end, we introduce a novel computational problem that we call Augmented LWE (A-LWE), differing from the original LWE problem only in the way the error term is produced. In fact, we embed arbitrary data into the error term without changing the target distributions. Following this, we prove that A-LWE instances are indistinguishable from LWE samples. This allows to build powerful encryption schemes on top of the A-LWE problem that are simple in its representations and efficient in practice while encrypting huge amounts of data realizing message expansion factors close to 1. This improves, to our knowledge, upon all existing encryption schemes. Due to the versatility of the error term, we further add various security features such as CCA and RCCA security or even plug lattice-based signatures into parts of the error term, thus providing an additional mechanism to authenticate encrypted data. Based on the methodology to embed arbitrary data into the error term while keeping the target distributions, we realize a novel CDT-like discrete Gaussian sampler that beats the best known samplers such as Knuth-Yao or the standard CDT sampler in terms of running time. At run time the table size amounting to 44 elements is constant for every discrete Gaussian parameter and the total space requirements are exactly as large as for the standard CDT sampler. Further results include a very efficient inversion algorithm for ring elements in special classes of cyclotomic rings. In fact, by use of the NTT it is possible to efficiently check for invertibility and deduce a representation of the corresponding unit group. Moreover, we generalize the LWE inversion algorithm for the trapdoor candidate of Micciancio and Peikert from power of two moduli to arbitrary composed integers using a different approach. \nIn the second part of this thesis, we present an efficient trapdoor construction for ideal lattices and an associated description of the GPV signature scheme. Furthermore, we improve the signing step using a different representation of the involved perturbation matrix leading to enhanced memory usage and running times. Subsequently, we introduce an advanced compression algorithm for GPV signatures, which previously suffered from huge signature sizes as a result of the construction or due to the requirement of the security proof. We circumvent this problem by introducing the notion of public and secret randomness for signatures. In particular, we generate the public portion of a signature from a short uniform random seed without violating the previous conditions. This concept is subsequently transferred to the multi-signer setting which increases the efficiency of the compression scheme in presence of multiple signers. Finally in this part, we propose the first lattice-based sequential aggregate signature scheme that enables a group of signers to sequentially generate an aggregate signature of reduced storage size such that the verifier is still able to check that each signer indeed signed a message. This approach is realized based on lattice-based trapdoor functions and has many application areas such as wireless sensor networks. \nIn the final part of this thesis, we extend the theoretical foundations of lattices and propose new representations of lattice problems by use of Cauchy integrals. Considering lattice points as simple poles of some complex functions allows to operate on lattice points via Cauchy integrals and its generalizations. For instance, we can deduce for the one-dimensional and two-dimensional case simple expressions for the number of lattice points inside a domain using trigonometric or elliptic functions.