Abstract The development of quantum computation enables exponential time complexity problems on classical computers to be solved in polynomial time on quantum computers. However, it also poses a threat to the security of classical cryptographic schemes based on integer factorization and discrete logarithms. In response to this challenge, quantum cryptographic schemes based on quantum computation and quantum communication environments have become a focal point of research. The quantum public-key cryptosystem based on the QSCDff problem stands as one of the influential schemes in the realm of quantum public-key cryptography, yet its feasibility remains unexplored in current literature. Our specific focus lies in the quantum circuit implementations and fault-tolerant construction, which serve as essential prerequisites for the physical feasibility of quantum cryptographic schemes. We provide quantum circuit implementations along with rigorous theoretical proofs for the computation of the permutation product operation and the permutation sign operation in quantum public-key cryptographic schemes. Based on the fault-tolerant quantum computation process of the aforementioned quantum circuit implementations, we propose two error-correction strategies and provide a theoretical feasibility analysis within a specified range in the ion-trap quantum computation environment, adhering to the theoretical limits of quantum computation. Rigorous proofs are presented to demonstrate the correctness and reliability of the proposed methods. Our contribution provides a theoretical foundation for the physical feasibility analysis of quantum cryptographic algorithms, offering insights into the challenges and prospects of implementing these algorithms in quantum computation environments.
The quantum circuit implementation and feasibility analysis of quantum public-key cryptosystem based on the QSCDff problem
Abstract The development of quantum computation enables exponential time complexity problems on classical computers to be solved in polynomial time on quantum computers. However, it also poses a threat to the security of classical cryptographic schemes based on integer factorization and discrete logarithms. In response to this challenge, quantum cryptographic schemes based on quantum computation and quantum communication environments have become a focal point of research. The quantum public-key cryptosystem based on the QSCDff problem stands as one of the influential schemes in the realm of quantum public-key cryptography, yet its feasibility remains unexplored in current literature. Our specific focus lies in the quantum circuit implementations and fault-tolerant construction, which serve as essential prerequisites for the physical feasibility of quantum cryptographic schemes. We provide quantum circuit implementations along with rigorous theoretical proofs for the computation of the permutation product operation and the permutation sign operation in quantum public-key cryptographic schemes. Based on the fault-tolerant quantum computation process of the aforementioned quantum circuit implementations, we propose two error-correction strategies and provide a theoretical feasibility analysis within a specified range in the ion-trap quantum computation environment, adhering to the theoretical limits of quantum computation. Rigorous proofs are presented to demonstrate the correctness and reliability of the proposed methods. Our contribution provides a theoretical foundation for the physical feasibility analysis of quantum cryptographic algorithms, offering insights into the challenges and prospects of implementing these algorithms in quantum computation environments.
Public-key cryptography is a key technology for making the Internet and other IT infrastructures secure. The security of the established public-key cryptosystems relies on the difficulty of factoring large composite integers or computing discrete logarithms. However, it is unclear whether these computational problems remain intractable in the future. For example, Shor showed in 1994 that quantum computers can be used to factor integers and to compute discrete logarithms in polynomial time. It is therefore necessary to develop alternative public-key cryptosystems which do not rely on the difficulty of factoring or computing discrete logarithms and which are secure even against quantum computer attacks. We call such cryptosystems quantum-immune. To prove the security of these quantum-immune cryptosystems, appropriate security models have to be used. Since quantum computers are able to solve problems in polynomial time which are supposed to be intractable for classical computers, the existing security models are inadequate in the presence of quantum adversaries. Therefore, new security models have to be developed to capture quantum adversaries. Properties of these new security models have to be investigated. On a more practical level, the quantum-immune cryptosystems have to be implemented in a way that they can seamlessly replace established cryptosystems. The implementations have to be efficient and suitable for resource-constrained devices. They must easily integrate into existing public-key infrastructures. This thesis contributes to both the theory and practice of quantum-immune cryptography, addressing the above-mentioned challenges. In the theoretical part, we concentrate on the quantum zero-knowledge property of interactive proof systems. We show for the first time that the quantum statistical, perfect, and computational zero-knowledge properties are preserved under sequential composition of interactive proof systems. In the practical part, we provide implementations of the most important quantum-immune cryptosystems. We present efficiency improvements of some of the alternative cryptosystems. The implementations are very efficient and easily integrate into existing public-key infrastructures. We present comprehensive timings that show that the alternative cryptosystems are competitive or even superior compared to established cryptosystems. Finally, we present a new cryptographic API that is particularly well-suited for resource-constrained devices like mobile phones and PDAs. With this API, the alternative cryptosystems can also be used with these devices.
Quantum algorithms can break factoring and discrete logarithm based cryptography and weaken symmetric cryptography and hash functions. In order to estimate the real-world impact of these attacks, apart from tracking the development of fault-tolerant quantum computers it is important to have an estimate of the resources needed to implement these quantum attacks. For attacking symmetric cryptography and hash functions, generic quantum attacks are substantially less powerful than they are for today's public-key cryptography. So security will degrade gradually as quantum computing resources increase. At present, there is a substantial resource overhead due to the cost of fault-tolerant quantum error correction. We provide estimates of this overhead using state-of-the-art methods in quantum fault-tolerance. We use state-of-the-art optimized circuits, though further improvements in their implementation would also reduce the resources needed to implement these attacks. To bound the potential impact of further circuit optimizations we provide cost estimates assuming trivial-cost implementations of these functions. These figures indicate the effective bit-strength of the various symmetric schemes and hash functions based on what we know today (and with various assumptions on the quantum hardware), and frame the various potential improvements that should continue to be tracked. As an example, we also look at the implications for Bitcoin's proof-of-work system. For many of the currently used asymmetric (public-key) cryptographic schemes based on RSA and elliptic curve discrete logarithms, we again provide cost estimates based on the latest advances in cryptanalysis, circuit compilation and quantum fault-tolerance theory. These allow, for example, a direct comparison of the quantum vulnerability of RSA and elliptic curve cryptography for a fixed classical bit strength.
Quantum algorithms can break factoring and discrete logarithm based cryptography and weaken symmetric cryptography and hash functions. In order to estimate the real-world impact of these attacks, apart from tracking the development of fault-tolerant quantum computers it is important to have an estimate of the resources needed to implement these quantum attacks. For attacking symmetric cryptography and hash functions, generic quantum attacks are substantially less powerful than they are for today's public-key cryptography. So security will degrade gradually as quantum computing resources increase. At present, there is a substantial resource overhead due to the cost of fault-tolerant quantum error correction. We provide estimates of this overhead using state-of-the-art methods in quantum fault-tolerance. We use state-of-the-art optimized circuits, though further improvements in their implementation would also reduce the resources needed to implement these attacks. To bound the potential impact of further circuit optimizations we provide cost estimates assuming trivial-cost implementations of these functions. These figures indicate the effective bit-strength of the various symmetric schemes and hash functions based on what we know today (and with various assumptions on the quantum hardware), and frame the various potential improvements that should continue to be tracked. As an example, we also look at the implications for Bitcoin's proof-of-work system. For many of the currently used asymmetric (public-key) cryptographic schemes based on RSA and elliptic curve discrete logarithms, we again provide cost estimates based on the latest advances in cryptanalysis, circuit compilation and quantum fault-tolerance theory. These allow, for example, a direct comparison of the quantum vulnerability of RSA and elliptic curve cryptography for a fixed classical bit strength.
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.
Quantum computing has advanced rapidly in recent decades, driven by developments across the technology stack, including quantum error-correcting codes and efficient quantum algorithms. Among these, quantum arithmetic circuits serve as fundamental building blocks for various promising algorithms. Despite their crucial role, the design of quantum arithmetic circuits faces challenges arising from the no-cloning theorem, qubit limitations, and circuit depth constraints, which significantly impact the efficiency of large-scale quantum computing. We provide an overview of quantum arithmetic circuits in the context of public-key cryptanalysis, with particular emphasis on optimization strategies such as measurement-based uncomputation and conditionally clean ancilla. We review state-of-the-art designs for essential arithmetic operations in public-key cryptanalysis such as addition, multiplication, and modular exponentiation. We also present an overview of the techniques used for fault-tolerant runtime and resource estimation in quantum cryptanalysis. In brief, this chapter emphasizes strategies for designing resource-efficient quantum arithmetic circuits, providing a basis for realistic evaluations of quantum cryptanalytic capabilities.
Quantum computing represents a transformative advancement in computational power, with profound implications for cryptographic systems. This paper explores the intersection of quantum computing and cryptography, examining the potential for quantum computers to both break traditional cryptographic algorithms and enable new forms of secure communication. Classical cryptographic systems, such as RSA and ECC, rely on the computational difficulty of problems like integer factorization and discrete logarithms. However, quantum algorithms, notably Shor's algorithm, can solve these problems exponentially faster than the best-known classical algorithms, posing a significant threat to the security of current cryptographic practices. This vulnerability necessitates the development and implementation of quantum-resistant algorithms to ensure data security in a post-quantum world. The paper reviews the state-of-the-art in quantum-resistant cryptographic algorithms, including lattice-based, hash-based, code-based, and multivariate polynomial cryptography. These algorithms are analyzed for their security properties, efficiency, and practicality for real-world deployment. Additionally, the paper discusses the ongoing standardization efforts led by organizations such as NIST, which are crucial for establishing widely accepted quantum-safe cryptographic protocols. Beyond the threat posed by quantum computing, this paper also highlights the potential benefits quantum computing can bring to cryptographic systems. Quantum Key Distribution (QKD) is one such advancement, offering theoretically unbreakable encryption by leveraging the principles of quantum mechanics. QKD enables secure communication channels that are immune to any computational advances, including those of quantum computers, by ensuring that any attempt at eavesdropping can be detected. The practical implementation challenges of QKD and other quantum cryptographic protocols are also addressed, including issues related to transmission distance, error rates, and integration with existing communication infrastructures. Experimental results from recent QKD trials are presented, showcasing significant progress in extending the viability of quantum-secure communication over long distances. Furthermore, the paper delves into the hybrid approaches combining classical and quantum cryptographic techniques. These approaches aim to leverage the strengths of both paradigms to enhance security and performance. For instance, hybrid protocols might use classical encryption for bulk data transfer, supplemented by quantum techniques for key exchange, thereby ensuring robust security even against future quantum adversaries. In conclusion, while quantum computing poses substantial risks to existing cryptographic systems, it also offers novel opportunities for creating highly secure communication protocols. The transition to quantum-safe cryptography is imperative to protect sensitive information in the quantum era. This paper underscores the urgency of advancing quantum-resistant cryptographic research, promoting international collaboration for standardization, and exploring innovative applications of quantum technologies in cryptography. Continued research and development in these areas are essential to stay ahead of the evolving threat landscape and harness the full potential of quantum computing for secure communications.