Shor’s algorithm [Shor 1994] is the main threat to classical public-key cryptography. Since its introduction in 1996, NTRU and its variants aim to develop cryptographic algorithms that are secure even against quantum computers. In this work, we study the matrix NTRU over integral domains proposed in 2023. We found that there is an error on the condition to avoid decryption failures and the key generation process is not practical due to severe limitations on matrix inversion. We propose a corrected statement for the decryption failure theorem and an expansion of the set of solutions when dealing with the problem of inverting matrix in Mn(Z[ √ −3]) that makes the key generation significantly faster.
Shor’s algorithm [Shor 1994] is the main threat to classical public-key cryptography. Since its introduction in 1996, NTRU and its variants aim to develop cryptographic algorithms that are secure even against quantum computers. In this work, we study the matrix NTRU over integral domains proposed in 2023. We found that there is an error on the condition to avoid decryption failures and the key generation process is not practical due to severe limitations on matrix inversion. We propose a corrected statement for the decryption failure theorem and an expansion of the set of solutions when dealing with the problem of inverting matrix in Mn(Z[ √ −3]) that makes the key generation significantly faster.
Most of the current public key cryptosystems’ security is based on the integer factorization and the discrete logarithm, which are mathematical problems that are considered difficult to solve. However, the publication in 1994 of the Shor’s algorithm [1] was a major setback for the world of cryptography, since it provided a quantum algorithm capable of solving efficiently these mathematical problems. Therefore, in recent years many governments and technology companies have put a lot of effort into developing what is known as post-quantum cryptography, i.e. the science that studies cryptosystems that are resistant to quantum attacks. Along these lines, in 2016 the National Institute of Standards & Technology launched a competition with the aim of establishing and standardising the best performing post-quantum cryptosystems, being the NTRU (Nth Degree Truncated polynomial Ring Units) cryptosystem one of the candidates still on the table. The aim of this project is to study the quantum-resistant cryptosystem NTRU, both from a theoretical and practical point of view. Firstly, the mathematical results on which it is based are presented and the existing attacks against it are analyzed. Finally, a practical implementation of the cryptosystem and four of its attack is carried out, from which experimental results are extracted.
Number theory research unit (NTRU) become the most important of securi-ty in recent, with its modification of their variant, this paper search of the literature and A number of studies have examined the in public key variant development and security. In general, prior work is limited to a subset of public key increasing complexity but the benefits of speed up encryption/ decryption have not been fully established. So this paper will be the basis for those who want to develop and find proposed solutions for new studies of the NTRU algorithm. This paper aims to develop a framework to investigate the NTRU development, had been discovered that despite its development over the years and even its acceptance in round three of post quantum cryptograph, then found that limit study in the new scope of quantum facility and the ability of hybrid of new study.
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.
Quantum computing has raised significant concerns regarding the security of classical cryptographic systems, particularly RSA, as it depends on the computational difficulty of factoring large semiprime numbers.In this study, we demonstrate the practical application of Shor's algorithm by successfully factorizing a semiprime integer, up to 30 bits (857830637 = 29167 X 29411) using the QuantumRings simulator.Our work highlights the effectiveness of Shor's algorithm in solving the factorization problem in polynomial time, which will significantly improve over the classical method for a large number.The results suggest that with advances in quantum hardware, such as increasing qubit counts and improving error correction, breaking larger RSA keys (e.g., RSA-2048) may soon become feasible.This seriously threatens current cryptographic systems, emphasizing the need to adopt post-quantum cryptography.Our findings aim to raise awareness among researchers, policymakers, and industry leaders of the importance of preparing for a quantum-safe future.
Public-key cryptography provides security for digital systems and communication. Traditional cryptographic solutions are constantly improved, e.g., to suppress brute-force attacks. However, Shor’s algorithm suited for quantum computers can break the bedrock of most currently used systems, i.e., the RSA problem and discrete logarithm problem. Post-quantum cryptography can withstand attacks carried out by quantum computers. Several families of post-quantum systems exist; one of them is isogeny-based cryptography. As a main contribution, in this paper, we provide a survey of chosen, fundamental isogeny-based schemes. The target audience of this review is researchers interested in practical aspects of this field of cryptography; therefore the survey contains exemplary implementations. Our goal was not to develop an efficient implementation, but to provide materials that make it easier to analyze isogeny-based cryptography.
The emergence of quantum computing is a serious threat to the principles of modern crypto-logical mathematics. The classical public-key cryptography, including Rivest-Shamir-Adleman (RSA), Elliptic Curve Cryptography (ECC), and Diffie-Hellman are very safe against classical adversaries yet become susceptible to quantum algorithms, including the Shor algorithm, which can easily solve the underlying mathematical problems. This weakness is jeopardizing confidentiality and integrity of the world-wide systems of communication, money transactions, and essential structures, and requires the immediate creation of safe alternatives. The current literature on post-quantum cryptography (PQC) has covered various classes of algorithm, such as lattice-based, code-based, multivariate schemes, and isogeny-based schemes. Although lattice-based approaches like Nth Degree Truncated Polynomial Ring Unit (NTRU) and FrodoKEM can be highly theoretically resistant, they can be computationally and memory-intensive. Cryptosystems based on code such as Classic McEliece are secure but not practicable because they have huge public key sizes. Multivariate schemes like Rainbow, and isogeny-based schemes like Super singular Isogeny Key Encapsulation (SIKE) have been attacked recently by cryptanalytic attacks, and broken completely. Such restrictions open the necessity to find more efficient and secure quantum-resistant solutions.This research study postulates a hybrid post-quantum cryptographic (PQC) system that combines CRYSTALS-Kyber for encryption with CRYSTALS-Dilithium for digital signatures both of which are approved through the National Institute of Standards and Technology (NIST) PQC standardization program. The model will be such that it ensures quantum resistance but at the same time it will be scalable, efficient and compatible with the existing infrastructures. The test outcomes indicate that the hybrid PQC model is 35-45 percent faster to encrypt and decrypt, signature performance is enhanced by up to 40 percent, and the throughput is 20 times higher (98.7 Mbps) than the same base PQC scheme including Nth Degree Truncated Polynomial Ring Unit (NTRU), McEliece and Rainbow. These enhancements affirm that the suggested model provides a viable trade-off between the security, speed, and efficiency in terms of key size, which makes it a viable and a sound base to ensure secure communication in the quantum age.