The rapid development of quantum computing poses challenges to the foundations of traditional cryptography. The threats are significant in terms of both asymmetric cryptography (which exposes schemes like RSA and ECC to efficient attacks) and symmetric cryptography, where key sizes must be increased to mitigate these threats. In this paper, we review the evolution of hash-based digital signatures, from early one-time signatures to modern stateless schemes, with an emphasis on their security properties, efficiency, and practical constraints. Moreover, we propose a simple comparative metric that reflects structural symmetry across key parameters such as key size, signature size, and computational cost, enabling a visual clustering of the schemes. We give particular attention to recent developments such as Verkle trees, which preserve symmetric design principles while improving scalability and proof compactness. The study highlights ongoing tradeoffs between stateful and stateless designs and argues for the continued relevance of symmetric cryptographic constructions in building secure, efficient post-quantum systems.
The rapid development of quantum computing poses challenges to the foundations of traditional cryptography. The threats are significant in terms of both asymmetric cryptography (which exposes schemes like RSA and ECC to efficient attacks) and symmetric cryptography, where key sizes must be increased to mitigate these threats. In this paper, we review the evolution of hash-based digital signatures, from early one-time signatures to modern stateless schemes, with an emphasis on their security properties, efficiency, and practical constraints. Moreover, we propose a simple comparative metric that reflects structural symmetry across key parameters such as key size, signature size, and computational cost, enabling a visual clustering of the schemes. We give particular attention to recent developments such as Verkle trees, which preserve symmetric design principles while improving scalability and proof compactness. The study highlights ongoing tradeoffs between stateful and stateless designs and argues for the continued relevance of symmetric cryptographic constructions in building secure, efficient post-quantum systems.
Modern public-key cryptography relies on the hardness of mathematical problems such as integer factorization and discrete logarithms. However, the development of quantum computing poses an imminent threat to these assumptions. Shor’s algorithm, in particular, can factor large semiprimes exponentially faster than classical algorithms, compromising systems like RSA, DSA, and ECC. This paper explores the mathematical foundations of pre-quantum cryptography, discusses the limitations of classical security models when confronted with quantum capabilities, and then pays attention to post-quantum cryptography (PQC), a field dedicated to developing cryptographic schemes resilient against both classical and quantum attacks. Among the proposed families, this paper focuses specifically on hash function–based cryptography for its simplicity and minimal reliance on algebraic structure. This study focuses in particular on SPHINCS+, a stateless hash-based digital signature scheme currently under consideration by NIST. Through detailed mathematical explanation and a visual example, we analyze its construction using Winternitz One-Time Signatures and Merkle trees. The results highlight SPHINCS+ as a robust candidate for post-quantum security due to its reliance on well-understood hash primitives and its resistance to known quantum algorithms such as Grover’s. Finally, this paper discusses ongoing challenges such as performance trade-offs, standardization, and real-world deployment. This research underscores the urgency of adopting quantum-resistant cryptographic systems before large-scale quantum computers become a reality.
Many public key cryptosystems in use today run the risk of being broken as quantum computing develops, affecting the security of popular commercial applications. Although there are possible defenses against quantum assaults, their complexity and inefficiency make them unsuitable for daily usage. This article discusses asymmetric hash-based digital signature algorithms, focusing on the symmetric and balanced structure achieved through the use of k-ary Verkle trees and lattice-based vector commitments. The study investigates the new concepts using vector commitments, a Verkle tree, and k-ary Verkle tree, which enhance efficiency and scalability while maintaining a symmetric structure in cryptographic proofs. Using cutting-edge Verkle tree technology, the authors of this paper provide a novel approach to creating a digital signature system. This is accomplished by using vector commitments, a k-ary Verkle tree, Verkle tree, and lattice-based vector commitments for post-quantum characteristics. The study also presents the ideas for designing post-quantum signatures using a k-ary Verkle tree, emphasizing the symmetry in tree structures and the balance between security and efficiency in asymmetric cryptographic systems.
Nowadays, quantum computing is developing at an unprecedented speed. This will pose a serious threat to the security of widely used public-key cryptosystems in the near future. Scientists are actively looking for ways to protect against quantum attacks; however, existing solutions still face different limitations in terms of efficiency and practicality. This paper explores hash-based digital signature schemes, post-quantum vector commitments and Verkle tree-based approaches for protecting against quantum attacks. The paper proposes an improved approach to generating digital signatures based on Verkle trees using lattice based vector commitments. In order to further reduce the memory space, the paper offers the methodology of integrating a post-quantum secure pseudo-random number generator into the scheme. Finally, the paper proposes an efficient post-quantum digital signature scheme based on Verkle trees, which minimizes memory requirements and reduces the signature size. Our proposed framework has strong resistance to quantum attacks, as well as high speed and efficiency. This study is an important contribution to the elaboration of post-quantum cryptosystems, which lays the foundation for developing secure and practical digital signature systems in the face of emerging quantum threats.
Quantum computing represents one of the most disruptive technological advancements of the 21st century, with profound implications for modern cryptographic systems. Traditional digital signature algorithms—such as RSA, DSA, and Elliptic Curve Cryptography (ECC)—derive their security from mathematical problems like integer factorization and discrete logarithms, which are computationally infeasible for classical computers. However, the advent of quantum algorithms, particularly Shor’s and Grover’s algorithms, poses a direct threat to these cryptographic assumptions by drastically reducing the time required to solve such problems. This development renders existing Digital Signature infrastructures potentially obsolete once large-scale quantum computers become operational. To counter this threat, researchers have proposed a range of post-quantum cryptographic (PQC) algorithms, including lattice-based, hash-based, and multivariate polynomial-based signature schemes, which are designed to resist quantum attacks. This paper critically examines the extent of quantum vulnerability in current Digital Signature systems and evaluates the comparative performance, efficiency, and security trade-offs of post-quantum alternatives. The study aims to provide a structured understanding of the ongoing transition toward quantum-resistant digital authentication and to propose a roadmap for secure implementation in future cryptographic environments.
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.
Modern-day cryptographic schemes are threatened by the rapid development of quantum computers, which are capable of solving discrete-logarithm-based problems in a reasonable amount of time. The paper proposes a novel approach to a PORS post-quantum secure signature. The main idea of the proposal is integrating a PORS signature scheme with a Verkle tree, which greatly reduces the signature size compared to a PORST that uses a Merkle tree, while reducing authentication overhead through constant-size vector commitment proofs at each tree level. The work is highly relevant as it proposes a post-quantum digital signature which has compact proofs and at the same time solves the problems of the original HORS signature scheme (subset-resilience problem). The work contributes to the ongoing standardization efforts in post-quantum cryptography by providing a theoretically scalable and efficient signature scheme that balances security, size, and verification speed.