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On the Theory and Practice of Quantum-Immune Cryptography

Martin Döring

Published Jan 1, 2008• 1 citation
Publisher

Abstract

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.

DOI
10.26083/tuprints-00001072

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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.
Public-key cryptography is indispensable in maintaining the security and integrity of digital data. The most widely used current public-key cryptography is based on the integer factorization problem and the elliptic-curve discrete logarithm problem, which are vulnerable against an adversary with large-scale quantum computers. Fortunately, post-quantum cryptography (PQC) can provide security against both classical and quantum adversaries. Due to rapid advancement in quantum computer development, the transition from classical public-key cryptography to PQC has become imperative. A watershed moment in this transition is the recent publication of a set of PQC schemes by the National Institute of Standards and Technology (NIST). Although it is a significant step, the research and development in PQC is quite immature compared to several decades-old classical public-key cryptographic schemes. Therefore, several open problems, such as physical attack analysis and their countermeasures, application-specific modifications, lightweight implementations for resource-constrained devices, integration into different secure protocols, etc., need to be addressed before the widespread deployment of PQC in real-world applications. This dissertation aims to address some of these problems in order to bridge the gap between the theory and practice of PQC.
Abstract. This paper presents a new paradigm of cryptography, quantum public-key cryptosystems. In quantum public-key cryptosystems, all parties including senders, receivers and adversaries are modeled as quantum (probabilistic) poly-time Turing (QPT) machines and only classical channels (i.e., no quantum channels) are employed. A quantum trapdoor one-way function, f, plays an essential role in our system, in which a QPT machine can compute f with high probability, any QPT machine can invert f with negligible probability, and a QPT machine with trapdoor data can invert f. This paper proposes a concrete scheme for quantum public-key cryptosystems: a quantum public-key encryption scheme or quantum trapdoor one-way function. The security of our schemes is based on the computational assumption (over QPT machines) that a class of subset-sum problems is intractable against any QPT machine. Our scheme is very efficient and practical if Shor’s discrete logarithm algorithm is efficiently realized on a quantum machine.
Cryptography is a critical technology in the modern computing industry, but the security of many cryptosystems relies on the difficulty of mathematical problems such as integer factorization and discrete logarithms. Large quantum computers can solve these problems efficiently, enabling the effective cryptanalysis of many common cryptosystems using such algorithms as Shor’s and Grover’s. If data integrity and security are to be preserved in the future, the algorithms that are vulnerable to quantum cryptanalytic techniques must be phased out in favor of quantum-proof cryptosystems. While quantum computer technology is still developing and is not yet capable of breaking commercial encryption, these steps can be taken immediately to ensure that the impending development of large quantum computers does not compromise sensitive data.
The security of current public-key cryptosystems relies on the hardness of factoring large integers or solving discrete logarithm problems. However, these mathematical problems can be solved in polynomial time using a quantum computer. This vulnerability has prompted research into post-quantum cryptography using alternative mathematical problems that are secure in the era of quantum computers. In this regard, the National Institute of Standards and Technology (NIST) began to standardize post-quantum cryptography in 2016. In this expository article, we give an overview of recent research on post-quantum cryptography. In particular, we describe the construction and security of multivariate polynomial cryptosystems and lattice-based cryptosystems, which are the main candidates of post-quantum cryptography.
Quantum computing poses a significant threat to conventional cryptographic systems that rely on the difficulty of mathematical problems such as integer factorization and discrete logarithms. These systems underpin much of the current security infrastructure, including public key cryptography and digital signatures. As quantum computers approach practical viability, there is an urgent need to transition to quantum-resistant cryptographic solutions that can secure digital communications against adversaries equipped with quantum capabilities. This paper explores the landscape of quantum-resistant cryptography, focusing on those algorithms that have emerged either as standards or as leading algorithm. Furthermore, the paper examines the progress of standardization efforts, such as the National Institute of Standards and Technology (NIST) Post-Quantum Cryptography (PQC) initiative, and the implications of deploying these algorithms in existing systems. By addressing the practical considerations for adoption, this study aims to provide a comprehensive overview of the current state and future directions of quantum-resistant cryptography, ensuring robust security in a post-quantum world.
The advent of quantum computing poses a significant existential threat to the security of contemporary public-key cryptographic systems, such as RSA and Elliptic Curve Cryptography (ECC), which underpin global digital communication and secure transactions. These systems rely on the computational hardness of problems like integer factorization and the discrete logarithm problem, which are efficiently solvable by Shor's algorithm on a sufficiently powerful quantum computer. This paper comprehensively rethinks the foundations of public-key cryptography by exploring ideal lattice-based schemes as a promising, quantum-secure alternative. We delve into the mathematical underpinnings of ideal lattices, particularly focusing on their compact representation and efficient arithmetic derived from polynomial rings. We elucidate how the conjectured hardness of ideal lattice problems, such as the Ring Learning With Errors (RLWE) problem and the Ideal Shortest Vector Problem (Ideal-SVP), provides robust security against both classical and quantum adversaries. Through a detailed analysis of their cryptographic properties, efficiency benefits, and security guarantees, we argue for the critical role of ideal lattices in forging the next generation of resilient public-key infrastructures. This research highlights the advantages of ideal lattices in terms of performance and security compactness, making them leading candidates in the ongoing standardization efforts for post-quantum cryptography.