Elliptic curves over Galois fields are widely used in modern cryptography. Cryptosystems based on elliptic curves are commonly deemed more secure than RSA for a given key size. However, with the rapid progress of quantum computing, the security of this traditional systems faces unprecedented challenge. To address this concern, this paper explores the resilience of a generalization of traditional elliptic curve cryptography. That is, we explore elliptic curves over non-prime rings (Zn), instead of fields. Elliptic curves over Zn for a composite integer n has been considered by researchers on information security. However, it is unclear how they stand against the unparalleled power of quantum computers. This article investigates quantum attacks on cryptosystems based on this new paradigm. The conclusion sheds light on the pressing and important task of searching for post-quantum cryptographic systems. In particular, the effectiveness of Shor’s algorithm (or its variation) on such systems is analyzed.
Elliptic curves over Galois fields are widely used in modern cryptography. Cryptosystems based on elliptic curves are commonly deemed more secure than RSA for a given key size. However, with the rapid progress of quantum computing, the security of this traditional systems faces unprecedented challenge. To address this concern, this paper explores the resilience of a generalization of traditional elliptic curve cryptography. That is, we explore elliptic curves over non-prime rings (Zn), instead of fields. Elliptic curves over Zn for a composite integer n has been considered by researchers on information security. However, it is unclear how they stand against the unparalleled power of quantum computers. This article investigates quantum attacks on cryptosystems based on this new paradigm. The conclusion sheds light on the pressing and important task of searching for post-quantum cryptographic systems. In particular, the effectiveness of Shor’s algorithm (or its variation) on such systems is analyzed.
As quantum computing progresses towards practical realization, the security of existing cryptographic protocols, such as RSA and Elliptic Curve Cryptography (ECC), faces unprecedented risks. Quantum algorithms like Shor's Algorithm threaten to undermine the security of these widely used encryption schemes, necessitating the development of quantum-resistant alternatives. This paper explores the vulnerabilities of current cryptographic protocols in the quantum era and investigates quantum-safe cryptographic techniques such as lattice-based, hash-based, and code-based cryptography. Additionally, we delve into Quantum Key Distribution (QKD), a method that leverages the principles of quantum mechanics to securely distribute cryptographic keys, ensuring secure communication even in the presence of quantum adversaries. The integration of QKD with post-quantum cryptographic algorithms is proposed as a robust approach to future-proofing cybersecurity. The paper concludes with a discussion of practical challenges in implementing quantum-safe systems and highlights future research directions to strengthen cryptographic resilience against quantum threats.
With the rapid advancement regarding quantum computing, traditional cryptographic systems face an existential threat because of their reliance concerning mathematical challenges which may be addressed effectively solved using quantum algorithms like the one developed by Shor. This imminent risk has driven extensive research into quantum-resistant cryptography (PQC), that seeks to develop encryption methods resilient against attacks powered by quantum computing. This study offers a thorough literature survey on high-level security techniques in quantum security for PQC, analysing past and current research efforts in this domain. We first examine the vulnerabilities of classical cryptographic protocols, including public-key encryption methods such as RSA (developed by Rivest, Shamir, and Adleman) and ECC (which uses elliptic curve mathematics), and traditional cryptographic hashing methods, against quantum adversaries. The study then explores the foundational approaches to quantum-secure cryptography, including lattice-based, hash-based, multivariate polynomial, code-based, and isogeny-based encryption methods that possess emerged as promising candidates for standardization. The survey highlights key challenges in the real-world application of PQC, including performance trade-offs, key size overhead, and deployment complexities. We conclude by identifying future research directions during the creation of scalable, efficient, and provably secure cryptographic methods designed to withstand quantum attacks techniques. As the shift toward the quantum-resilient era approaches, this survey aims to provide a clear and structured comprehension of the evolving the domain of quantum security for PQC.
The rapid advancements in quantum computation pose significant threats to traditional cryptographic systems, such as those used in the Internet of Things (IoT). RSA and methods like Elliptic Curve Cryptography (ECC) are particularly vulnerable to quantum algorithmic techniques, such as Shor’s and Grover’s algorithms. This paper focuses on the analysis of cryptographic models based on quantum technologies, with special emphasis on IoT security. It examines how both symmetric and public-key cryptography are affected by quantum comput-ing—highlighting vulnerabilities in symmetric schemes to Grover’s algorithm and in public-key systems like ECC to Shor’s algorithm. Significant focus is placed on the advancement of quantum-resistant ECC, with an in-depth evaluation of its effectiveness, security capabilities, and suitability for IoT applications. This work also compares various methods for implementing quantum-enhanced ECC, considering both software and hardware solutions. Furthermore, the paper explores emerging hybrid systems that integrate classical and quantum-resistant encryption to offer enhanced security in transitional scenarios. Lastly, it outlines potential research avenues for developing new techniques to protect future IoT networks from quantum-based attacks.
The advent of quantum computing poses a significant threat to the security of current asymmetric cryptographic systems, which underpin much of modern digital communication and security. These systems, predominantly RSA and Elliptic Curve Cryptography (ECC), rely on the computational hardness of mathematical problems such as integer factorization and the discrete logarithm problem. Shor's algorithm, a quantum algorithm, demonstrates the ability to efficiently solve these problems, rendering classical public-key cryptography vulnerable to attacks by sufficiently powerful quantum computers. This paper argues for and thoroughly examines the strategic imperative of transitioning from these classical cryptographic foundations to a new paradigm based on lattices, which are believed to be resistant to quantum attacks. We delve into the fundamental mathematical problems underlying lattice-based cryptography, such as the Shortest Vector Problem (SVP) and the Learning With Errors (LWE) problem, whose worst-case hardness is leveraged to achieve cryptographic security. A comprehensive literature review is provided, tracing the evolution of quantum threats and the emergence of lattice-based schemes. The methodology details the construction principles of prominent lattice-based cryptosystems. Results demonstrate their theoretical security against quantum adversaries and highlight their practical attributes, including compact key sizes and competitive performance. The discussion section contextualizes lattice-based cryptography within the broader landscape of post-quantum candidates, examining their strengths, weaknesses, and the ongoing standardization efforts by organizations like NIST. We conclude that lattice-based cryptography offers a robust and promising foundation for building resilient asymmetric cryptographic systems in the post-quantum era, ensuring the long-term security of digital information through a necessary re-architecture of cryptographic primitives.
This article analyzes the threats to the security of modern cryptographic systems posed by the development of quantum computers, as well as the emergence of the concept of post-quantum cryptography as a solution to these problems. The study demonstrates that classical public-key cryptosystems, particularly RSA, DSA, and elliptic curve-based algorithms, can be broken using quantum algorithms. At the same time, new cryptographic systems based on hash functions, codes, lattices, and multivariate quadratic equations are considered a promising direction. The main challenges related to the efficiency, reliability, and applicability of post-quantum cryptography are also highlighted. As a result, the necessity of developing new cryptographic standards to ensure future information security is demonstrated.
The boom of quantum computing has brought both benefits and threats to cyber-physical systems. The primary purpose of the paper is to highlight the possible threats by quantum computers, and to provide a comparative study of the on-going quantum-resistant post-quantum cryptography suitable for current network infrastructure. Asymmetric key cryptography, mainly the widely adopted RSA and ECC algorithms, has been compromised by quantum algorithms such as the Shor’s algorithm. Moreover, symmetric algorithms such as AES are weakened by brute force attacks using Grover’s algorithm. Thus, researchers are developing public key algorithms based on hard mathematical models that provide resistance against quantum attacks and can be deployed in classical hardware. We categorize the well-known post-quantum schemes based on (1) their security goals: confidentiality and integrity, and (2) their mathematical problems: code-based, lattice-based, super-singular elliptic curve isogeny based, multivariate and hash-based algorithm. Hash-based post- quantum cryptography is further categorized into two: stateless and stateful signature scheme. Furthermore, the chapter compares the algorithms based on the key sizes, ciphertext sizes, signature sizes, computation, and time cost. The paper also presents the challenges faced by the post-quantum algorithms, and it outlines directions for further research to improve the algorithms. Moreover, various cyber-physical fields, including industrial control systems, mainly nuclear power plants, water management plants, and even space stations and their infrastructure, rely on classical cryptography. Thus, the paper presents a generic hybrid cryptosystem based on classical and post-quantum cryptography to enhance resistance against classical and quantum attacks.