Post-Quantum Cryptosystems: Open Problems and Solutions. Lattice-Based Cryptosystems
Ekaterina Sergeevna Malygina, A. V. Kutsenko, S. A. Novoselov, N. S. Kolesnikov, Alexander Olegovich Bakharev, Хильчук Ирина Сергеевна, A. S. Shaporenko, Natalia Tokareva
Abstract The paper provides an overview of the main approaches to the construction of post-quantum cryptographic systems that are currently used. The area of lattice-based cryptography is analyzed in detail. We give the description and characterization of some known lattice-based cryptosystems whose resilience is based on the complexity of the shortest vector problem, learning with errors problem, and their variations. The main approaches to solving the problems from lattice theory, on which attacks on the corresponding cryptosystems are based, are analyzed. In particular, some known theoretical estimates of time and memory complexity of lattice basis reduction and lattice sieving algorithms are presented.
Abstract The paper provides an overview of the main approaches to the construction of post-quantum cryptographic systems that are currently used. The area of lattice-based cryptography is analyzed in detail. We give the description and characterization of some known lattice-based cryptosystems whose resilience is based on the complexity of the shortest vector problem, learning with errors problem, and their variations. The main approaches to solving the problems from lattice theory, on which attacks on the corresponding cryptosystems are based, are analyzed. In particular, some known theoretical estimates of time and memory complexity of lattice basis reduction and lattice sieving algorithms are presented.
The landscape of modern cryptography, traditionally anchored in the computational hardness of number-theoretic problems such as integer factorization and the discrete logarithm problem, faces an existential threat from the advent of quantum computing. Shor's algorithm, discovered in 1994, demonstrates the theoretical capability of a sufficiently powerful quantum computer to efficiently solve these problems, rendering widely used cryptographic schemes like RSA and Elliptic Curve Cryptography (ECC) insecure. This imminent threat necessitates a paradigm shift towards post-quantum cryptography (PQC), which aims to develop cryptographic algorithms resistant to both classical and quantum attacks. Among the various candidates for PQC, lattice-based cryptography has emerged as a particularly promising area. This paper provides a comprehensive overview of the transition from classical number theory-based cryptography to lattice-based schemes. We explore the fundamental mathematical principles underlying lattices, introduce key lattice problems such as the Shortest Vector Problem (SVP), Closest Vector Problem (CVP), Small Integer Solution (SIS), and Learning With Errors (LWE), and discuss their application in constructing quantum-resistant cryptographic primitives. Furthermore, we analyze the security, efficiency, and practical considerations of prominent lattice-based cryptosystems, highlighting their role in the ongoing NIST post-quantum cryptography standardization process. The paper concludes by discussing the challenges and future directions in the deployment and research of lattice-based cryptographic solutions.
This thesis explores key literature on lattice-based post-quantum cryptography and its elements. Chapter 2 introduces key computational concepts, including complexity theory and various models of computation, to establish the theoretical basis for cryptographic security. Chapter 3 provides an overview of classical cryptographic techniques, setting the stage for the transition to quantum-resistant approaches. Chapters 4 and 5 investigate lattice theory and its associated hard problems. Chapter 6 focuses on classical algorithms for lattice reduction that are relevant to both analyzing and solving lattice-based cryptographic problems. Appendices A and B supplement the main content with relevant mathematical back- ground and introductory concepts in quantum computing.
The landscape of modern cryptography is fundamentally challenged by the advent of quantum computing. Current public-key cryptosystems, which form the bedrock of secure digital communication, rely on the computational hardness of problems such as integer factorization (RSA) and the discrete logarithm problem (ECC). However, Shor's algorithm, if implemented on a sufficiently powerful quantum computer, can efficiently break these schemes, rendering them insecure. This paper explores the critical transition from these classical number theory problems to lattice-based cryptography, presenting a comprehensive analysis of the motivations, theoretical underpinnings, and practical implications for forging post-quantum cryptographic security. We delve into the mathematical foundations of lattice problems, such as the Shortest Vector Problem (SVP) and Learning With Errors (LWE), which are believed to resist known quantum algorithms. Through a thorough literature review, we trace the evolution of cryptographic thought, detailing the vulnerabilities of pre-quantum schemes and the emergence of lattice-based constructions as leading candidates for post-quantum standardization. The methodology outlines a comparative theoretical analysis of security, efficiency, and robustness against both classical and quantum attacks. Results demonstrate the compelling advantages of lattice-based schemes in terms of provable security reductions and resistance to quantum threats, albeit with considerations for increased key and ciphertext sizes. The discussion highlights the challenges and opportunities in the deployment and standardization of these new cryptographic primitives, emphasizing their critical role in securing future digital infrastructures. This work concludes that lattice-based cryptography offers a robust and theoretically sound pathway to establishing enduring security in a quantum-threatened world.
The advent of large-scale quantum computers poses an existential threat to the security of current public-key cryptosystems, such as RSA and Elliptic Curve Cryptography (ECC), which rely on the computational intractability of integer factorization and discrete logarithms. This paper explores lattice-based cryptography as a promising candidate for post-quantum security. We delve into the mathematical foundations of lattices, specifically focusing on hard problems like the Shortest Vector Problem (SVP), Closest Vector Problem (CVP), Small Integer Solution (SIS), and Learning With Errors (LWE) problems, which are believed to remain intractable even for quantum computers. We provide a comprehensive review of prominent lattice-based constructions, including NTRU, Kyber, and Dilithium, highlighting their security proofs, efficiency considerations, and suitability for various cryptographic primitives such as public-key encryption, key-exchange mechanisms, and digital signatures. Furthermore, we discuss the challenges and opportunities associated with their deployment, including parameter selection, resistance to side-channel attacks, and integration into existing cryptographic infrastructures. By examining the theoretical underpinnings and practical implications, this work aims to solidify understanding of lattice-based methods as a robust and essential component of the cryptographic landscape in the quantum era, moving beyond the traditional RSA frontier.
The advent of large-scale quantum computers poses a profound and imminent threat to the security foundations of modern cryptography. Widely deployed public-key cryptosystems, such as RSA and Elliptic Curve Cryptography (ECC), rely on problems like integer factorization and discrete logarithms, which Shor's algorithm can efficiently solve. This necessitates a proactive transition to post-quantum cryptography (PQC). Among leading candidates, lattice-based cryptography stands out due to its robust security guarantees, based on hard problems lacking known efficient quantum solutions, and structural properties facilitating efficient implementation. This paper explores the fundamental principles of lattice-based cryptography, detailing underlying mathematical hard problems such as the Shortest Vector Problem (SVP) and Learning With Errors (LWE). We review the evolution of lattice-based schemes, from early constructions to contemporary candidates like Kyber and Dilithium, emphasizing their suitability for Key Encapsulation Mechanisms (KEMs) and Digital Signature Schemes (DSS). Furthermore, we analyze their performance characteristics, security assumptions, and resilience against classical and quantum attacks, providing a critical comparison with other PQC families. This comprehensive overview, structured through a systematic critical analysis framework, highlights the ascendancy of lattice-based cryptography as a cornerstone for future quantum-resistant security infrastructure, offering theoretical underpinnings, practical implications, and actionable future research directions for a post-quantum world.
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.