Lattice-based cryptosystems rely on certain lattice problems for their socurity, such as the shortest vector and the closest vector probtems. NTRU is a kind of lattice based public key cryptosysytem gaining much attention because of its high efficiency. Its encryption and decryption use a mixing system suggested by polynomial algebra combined with a clustering principle based on elementary probability theory. With the availability of quantum computers, the security of cryptosystems based on classical mathematic problems needs to be reevaluated. Since lattice reduction is the main threat to lattice-based cryptosystems, lattice reduction using quantum search algorithms are analyzed to evaluate the security of NTRU in this paper. According to our results, original security paratneters proposed for NTRU should be increased in the event that Grover's quantum search algorithm is used for lattice reduction.
Lattice-based cryptosystems rely on certain lattice problems for their socurity, such as the shortest vector and the closest vector probtems. NTRU is a kind of lattice based public key cryptosysytem gaining much attention because of its high efficiency. Its encryption and decryption use a mixing system suggested by polynomial algebra combined with a clustering principle based on elementary probability theory. With the availability of quantum computers, the security of cryptosystems based on classical mathematic problems needs to be reevaluated. Since lattice reduction is the main threat to lattice-based cryptosystems, lattice reduction using quantum search algorithms are analyzed to evaluate the security of NTRU in this paper. According to our results, original security paratneters proposed for NTRU should be increased in the event that Grover's quantum search algorithm is used for lattice reduction.
In this paper we report a more secure and efficient encryption algorithm based on the NTRU cryptographic scheme.NTRU is lattice based scheme resistant to quantum computing, hence it falls under the class of post quantum cryptosystems.It is based on shortest vector problem (svp) [9].The main characteristics of the system are low memory and low computational requirements but it provides high level of security.We present modifications in the NTRU scheme for making it more secure and efficient particul arly for applications in wireless and constrained devices.In the original scheme, repetitions in the plaintext message lead to repetitions in the cipher text, which is a source of weakness in the system.To overcome this problem each byte of the input has been digested with different operations that produce different encrypted text even for repeated content of the Plain text message.The second modification is enhancing the public key scheme that makes this system more robust.These two modifications in the NTRU scheme makes it secure even for use in the Quantum Computing environment.
This chapter introduces lattice-based cryptography, a critical mathematical framework for securing communications in the post-quantum era. While classical public-key cryptosystems rely on the intractability of integer factorization and discrete logarithms, these foundations are fundamentally vulnerable to quantum computing threats, specifically Shor's algorithm. To address this vulnerability, the text explores cryptosystems built upon the algebraic structure of lattices, defined as discrete points in n-dimensional Euclidean space generated by linearly independent vectors. The chapter systematically covers the mathematical properties of lattices, associated computational problems, and distance metrics. It provides an in-depth examination of lattice basis reduction algorithms, which are fundamental to both the design and security evaluation of these systems. Furthermore, the chapter details the operational mechanics—including encryption, decryption, and cryptanalysis—of prominent post-quantum architectures, specifically the GGH and NTRU public-key cryptosystems. By transitioning from classical number theory to high-dimensional geometry, this chapter equips readers with the cryptographic primitives necessary to withstand future quantum attacks.
The advent of quantum computing poses a significant existential threat to the security foundations of current public-key cryptography, most notably the widely adopted RSA algorithm. Shor's algorithm, specifically, can efficiently break the mathematical problems underlying RSA and elliptic curve cryptography, necessitating a paradigm shift towards quantum-resistant solutions. This paper explores lattice-based cryptography as the leading candidate for this post-quantum era. We delve into the fundamental mathematical principles of lattices, including hard problems like the Shortest Vector Problem (SVP), Closest Vector Problem (CVP), Short Integer Solution (SIS), and Learning With Errors (LWE), which form the bedrock of their security. The paper reviews prominent lattice-based schemes such as NTRU, CRYSTALS-Kyber, CRYSTALS-Dilithium, and Falcon, highlighting their construction, security properties, and performance characteristics. We analyze their resilience against both classical and quantum attacks, discuss their progress in the NIST Post-Quantum Cryptography Standardization process, and evaluate their practical implications and implementation challenges. The discussion emphasizes the provable security, versatility, and efficiency of lattice-based methods, positioning them as the robust and essential successor to RSA for safeguarding digital communications and data in a quantum-enabled future.
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.
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.
The safety and security of various cryptosystems are related to the hardness of the integer factorization problem and discrete logarithm problem, most cryptosystems will be rendered vulnerable to attack and outdated as quantum computers become more advanced.This study examines lattice-based cryptography (the NTRU cryptosystem), compares the suggested variation to other cryptosystems by simulation and presents the execution time for key generation, encryption, and decryption procedures, as well as to highlight the benefits and drawbacks of the proposed cryptosystem.A variant of the NTRU cryptosystem is developed with the purpose of improving the cryptosystem's security by allowing it to withstand attacks in both classical and quantum computing.The simulation revealed that the existing NTRU cryptosystem has a better time complexity compared to the proposed NTRU cryptosystem.The existing NTRU cryptosystem has a time complexity of O(n 2 ) while the proposed variant has O(n 3 ).Though had a better capacity of handling large data because of the introduction of lattice multiplication.