Journal of Discrete Mathematical Sciences and Cryptography
A comparative analysis of post-quantum cryptographic schemes over discrete algebraic structures for enhanced information security in large-scale networks
Abstractly, the fast development of quantum computing is a threat to classical public-key cryptographic infrastructures.Asymmetric algorithms (which are widely used and include RSA and Elliptic Curve Cryptography (ECC)) are susceptible to the Shor algorithm, which can break them in a time-polynomial manner.This paper will give a detailed comparative study of Post-Quantum Cryptography (PQC) schemes founded on discrete algebraic structures such as lattice-based, code-based, hash-based, and multivariate polynomials based cryptosystems.These paradigms are systematically assessed according to computational complexity, key and ciphertext sizes, security reduction assumptions, and how they can be deployed on large scale networks.The NIST-standardized algorithms, which are the focus of our analysis, are ML-KEM (CRYSTALS-Kyber), ML-DSA (CRYSTALS Dilithium), FN-DSA (Falcon), and SLH-DSA (SPHINCS+).We also consider hybrid cryptography transitional measures to provide backward compatibility and security guarantee in the process of migrating the classical to quantum-resistant systems.As we found, the current state of lattice-based schemes provides the most desirable ratio of security to performance in large-scale applications, although, in the long run, code-based and multivariate schemes are necessary as their complements.
Abstractly, the fast development of quantum computing is a threat to classical public-key cryptographic infrastructures.Asymmetric algorithms (which are widely used and include RSA and Elliptic Curve Cryptography (ECC)) are susceptible to the Shor algorithm, which can break them in a time-polynomial manner.This paper will give a detailed comparative study of Post-Quantum Cryptography (PQC) schemes founded on discrete algebraic structures such as lattice-based, code-based, hash-based, and multivariate polynomials based cryptosystems.These paradigms are systematically assessed according to computational complexity, key and ciphertext sizes, security reduction assumptions, and how they can be deployed on large scale networks.The NIST-standardized algorithms, which are the focus of our analysis, are ML-KEM (CRYSTALS-Kyber), ML-DSA (CRYSTALS Dilithium), FN-DSA (Falcon), and SLH-DSA (SPHINCS+).We also consider hybrid cryptography transitional measures to provide backward compatibility and security guarantee in the process of migrating the classical to quantum-resistant systems.As we found, the current state of lattice-based schemes provides the most desirable ratio of security to performance in large-scale applications, although, in the long run, code-based and multivariate schemes are necessary as their complements.
The advent of large-scale quantum computers poses an existential threat to the cryptographic algorithms that underpin modern digital security. Shor's algorithm enables quantum computers to factor large integers and compute discrete logarithms in polynomial time, rendering RSA, Elliptic Curve Cryptography (ECC), and Diffie-Hellman key exchange fundamentally insecure. Meanwhile, Grover's algorithm provides a quadratic speedup for brute-force search, effectively halving the security strength of symmetric ciphers and hash functions. This paper presents a comprehensive investigation into the impact of quantum computing on both asymmetric and symmetric cryptographic algorithms. We analyze the computational resources required by quantum attacks on current standards, evaluate the timeline of quantum threats based on hardware roadmaps from leading quantum computing companies, and conduct a systematic performance comparison of NIST-standardized post-quantum cryptographic (PQC) algorithms. Our experimental evaluation covers key generation time, encapsulation/decapsulation latency, signature generation/verification speed, key sizes, and bandwidth overhead across lattice-based, code-based, hash-based, and multivariate cryptographic schemes. Results indicate that lattice-based schemes, particularly CRYSTALS-Kyber and CRYSTALS-Dilithium, offer the most favorable balance of security and performance for practical deployment. We also propose a phased migration framework for transitioning existing cryptographic infrastructure to quantum-resistant alternatives, addressing compatibility, performance, and operational challenges.
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.
The advent of quantum computing poses a significant challenge to conventional cryptographic methods such as RSA, Elliptic Curve Cryptography (ECC), and Diffie-Hellman key exchange. Quantum algorithms, particularly Shor’s algorithm, have the potential to break these encryption techniques, making it essential to develop cryptographic approaches that can withstand quantum threats. Post-quantum cryptography (PQC) has emerged as a crucial area of research, aiming to establish cryptographic mechanisms that remain secure even in the presence of quantum adversaries. This study presents a detailed comparative analysis of five primary categories of quantum-resistant cryptographic algorithms: lattice-based, code-based, hash-based, multivariate polynomial, and isogeny-based encryption schemes. Each of these approaches offers distinct advantages and challenges in terms of security, efficiency, and implementation feasibility. Among them, lattice-based cryptography has gained significant attention due to its robust security properties and computational efficiency, making it a strong candidate for standardization. Conversely, code-based cryptography provides high security but is hindered by its large key sizes, affecting its practical deployment. The research includes a real-time performance assessment of selected PQC algorithms, analyzing key factors such as encryption speed, key size, and computational demands. Furthermore, the study examines the challenges associated with transitioning from classical encryption standards to quantum-resistant frameworks, including compatibility constraints, computational overhead, and the necessity for global standardization. Potential mitigation approaches, such as hybrid cryptographic techniques that integrate both classical and post-quantum encryption models, are also explored. Our findings emphasize that while quantum-resistant cryptography is still evolving, early adoption of PQC frameworks is essential for safeguarding future communication networks. This paper provides valuable insights for researchers, cybersecurity professionals, and policymakers on strategic measures to implement quantum-secure encryption systems effectively.
Not only is quantum computing a game changer but it is also an unprecedented threat to world security in cyberspace. Although quantum systems are expected to transform scientific computing, they are also endangering the very notion of classical cryptography, specifically, the public key system of RSA and elliptic curve cryptography (ECC). The problems in computation that such schemes are built on, including integer factorization and discrete logarithms, can be efficiently solved using quantum algorithms, including Shor. This makes the secured communication network by classical encryption susceptible to quantum attacks in the future. In turn, current quantum-resistant or post-quantum cryptographic algorithms (PQC) are resistant to classical and quantum adversaries. The article provides a general overview of significant families of PQC, such as lattice-based, code-based, multivariate polynomial, hash-based and isogeny-based, and discusses their relevance to the provisioning of communication networks. The work illuminates to some degree the advances toward standardization already being undertaken by the National Institute of Standards and Technology (NIST) by analysing its security underpinnings, trade-offs in its operation and its readiness to deploy. Experiments show that lattice based (CRYSTALS-Kyber, Dilithium) and code based (McEliece) are both well-theoretically secure as well as high performance, and that systems based on code (McEliece) have a long track record of reliability at the cost of large key-sizes. Other issues which have been talked about in the paper are interoperability issues, migration policies and side-channel attack defence. It finds conclusively that quantum-resistant cryptography is the next step that must be regarded as the most important to secure the secrecy and integrity of communication networks in the future.
As quantum computing advances, current cryptographic protocols are increasingly vulnerable to quantum attacks, particularly those based on Public Key Infrastructure (PKI) like RSA or Elliptic Curve Cryptography (ECC). This paper presents a comprehensive review of Post-Quantum Cryptography (PQC) as a solution to protect digital systems in the quantum era. We provide an in-depth analysis of various quantum-resistant cryptographic algorithms, including lattice-based, code-based, hash-based, isogeny-based, and multivariate approaches. The review highlights the National Institute of Standards and Technology (NIST) PQC standardization process, highlighting key algorithms, such as CRYSTALS–Kyber, CRYSTALS–Dilithium, Falcon, and SPHINCS+, and discusses the strengths, vulnerabilities, and implementation challenges of the leading algorithms. In addition, we explore transition strategies for organizations, emphasizing hybrid cryptography to ensure backward compatibility during migration. This study offers key insights into the future of cryptographic standards and the critical steps necessary to prepare for the transition from classical to quantum-resistant systems.
The advance of quantum computing poses significant risks to traditional cryptographic systems, including RSA and ECC, due to algorithms such as Shor's, prompting an urgent need for robust post-quantum cryptography (PQC) solutions. This paper presents a systematic review of PQC algorithms, encompassing lattice-based, code-based, multivariate polynomial, hash-based, and supersingular isogeny-based cryptography. By evaluating recent publications and NIST standardization efforts, we assess these approaches in terms of quantum resistance, computational efficiency, scalability, and implementation challenges. Lattice-based schemes, notably CRYSTALS-Kyber, excel in quantum-secure encryption and performance, whereas codebased systems like McEliece are hindered by large key sizes. Hash-based signatures, such as SPHINCS+, and isogeny-based protocols, including SIDH, provide promising alternatives but require further optimization to reduce computational overhead. Our analysis highlights the potential of hybrid PQC methods and lightweight hardware implementations for securing emerging domains like IoT, blockchain, and financial systems. The review serves as a foundational resource for researchers and practitioners, advocating the adoption of quantum-resistant cryptography to preserve data confidentiality, integrity, and authenticity in the quantum era.