Abstract The emerging technologies like quantum computing and cloud computing have a complex computing power and can break the security of the classical cryptographic constructions (like ElGamal's cryptosystem). Thus, we were motivated to work on hard lattice problems, which are more complex and can resist these new modern technologies. In this proposal, we present a lattice‐based version of discrete logarithm problem‐based ElGamal public‐key encryption and signature schemes that exhibit strong security features and efficient implementation. The security of these presented schemes is based on the worst‐case hardness of approximating a small integer vector in their corresponding lattice. Furthermore, this proposed work illustrates a security proof of the proposed schemes and shows that the presented schemes are well protected in the modern computing environment. The performance analysis and comparison with classical ElGamal schemes and recent lattice‐based schemes are also made, and it is seen that the proposed schemes achieve better performance.
Abstract The emerging technologies like quantum computing and cloud computing have a complex computing power and can break the security of the classical cryptographic constructions (like ElGamal's cryptosystem). Thus, we were motivated to work on hard lattice problems, which are more complex and can resist these new modern technologies. In this proposal, we present a lattice‐based version of discrete logarithm problem‐based ElGamal public‐key encryption and signature schemes that exhibit strong security features and efficient implementation. The security of these presented schemes is based on the worst‐case hardness of approximating a small integer vector in their corresponding lattice. Furthermore, this proposed work illustrates a security proof of the proposed schemes and shows that the presented schemes are well protected in the modern computing environment. The performance analysis and comparison with classical ElGamal schemes and recent lattice‐based schemes are also made, and it is seen that the proposed schemes achieve better performance.
Modern technologies like quantum and cloud computing have reshaped computing, offering immense power and scalability. While beneficial, they also challenge the security of traditional cryptographic systems. Quantum computing threatens RSA and ECC with algorithms like Shor’s algorithm, which can accelerate computations exponentially. This risks exposing these systems to attacks, necessitating quantum-resistant cryptography. Cloud computing poses data security concerns, requiring robust cryptographic mechanisms and access controls. Lattice-based cryptography, leveraging problems like the Short Integer Solution (SIS), emerges as a solution. This paper presents a novel quantum-resistant public key encryption scheme based on ElGamal and SIS, ensuring security against quantum and classical threats in modern cryptographic environments.
Why study Lattice-based Cryptography? There are a few ways to answer this question. 1. It is useful to have cryptosystems that are based on a variety of hard computational problems so the different cryptosystems are not all vulnerable in the same way. 2. The computational aspects of lattice-based cryptosystem are usually simple to understand and fairly easy to implement in practice. 3. Lattice-based cryptosystems have lower encryption/decryption computational complexities compared to popular cryptosystems that are based on the integer factorisation or the discrete logarithm problems. 4. Lattice-based cryptosystems enjoy strong worst-case hardness security proofs based on approximate versions of known NP-hard lattice problems. 5. Lattice-based cryptosystems are believed to be good candidates for post-quantum cryptography, since there are currently no known quantum algorithms for solving lattice problems that perform significantly better than the best-known classical (non-quantum) algorithms, unlike for integer factorisation and (elliptic curve) discrete logarithm problems. 6. Last but not least, interesting structures in lattice problems have led to significant advances in Homomorphic Encryption, a new research area with wide-ranging applications.
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.
Abstract Almost all practically relevant asymmetric cryptosystems like RSA or ECC are either based on the hardness of factoring or on the hardness of the discrete logarithm problem. However, both problems could be solved efficiently on a large enough quantum computer. While quantum computers powerful enough to break currently used parameter sets are not available yet, they are heavily researched and expected to reach maturity in 15 to 20 years. As a consequence, research on alternative quantum-safe cryptosystems is required. One alternative is lattice-based cryptography which allows the construction of asymmetric public-key encryption and signature schemes that offer a good balance between security, performance, and key as well as ciphertext sizes.
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.
The paper offers a novel digital signature scheme that integrates ElGamal cryptographic principles with the Short Integer Solution (SIS) problem, specifically designed to ensure post-quantum security. As quantum computers advance and present significant risks to traditional cryptographic systems, this scheme offers an interesting alternative for securing digital signatures against potential quantum threats. The scheme uses only basic secure principles. The offered approach offers key generation, where parameters and random matrices are selected, and signature generation, which involves creating signatures based on hashed messages and matrix computations. Verification ensures the authenticity and integrity of signatures. We provide experimental evaluations detailing key generation, signature creation, and verification times across different matrix dimensions and message sizes. Key generation takes between 2.5–10.2 seconds, while signature generation ranges from 0.20 to 9.30 milliseconds and verification from 0.18 to 8.90 milliseconds, depending on message size and matrix dimension. The scheme maintains a consistent signature size of 1.7 KB, independent of message length due to a hash-and-sign strategy. These results demonstrate that the scheme balances post-quantum security with practical performance, especially in high-security contexts. A comparison with traditional ElGamal encryption reveals the trade-offs between security and efficiency. While the SIS-based scheme delivers enhanced protection against quantum threats, it also entails increased computational complexity and larger signature sizes compared to conventional schemes. Overall, our proposed digital signature scheme stands as an excellent option for safe communications in a post-quantum world, representing a crucial step in protecting the authenticity and integrity of digital exchanges against changing technological risks. We believe that as quantum computing continues to develop, research into robust cryptographic alternatives will become increasingly important for safeguarding sensitive information across various sectors.