Abstract. Lattice-based cryptography is one of the candidates in the area of post-quantum cryptography. Cryptographic schemes with security reductions to hard lattice problems (like the Shortest Vector Problem SVP) offer an alternative to recent number theory-based schemes. In order to guarantee asymptotic efficiency, most lattice-based schemes are instantiated using polynomial rings over integers. These lattices are called ideal lattices. It is assumed that the hardness of lattice problems in lattices over integer rings remains the same as in regular lattices. In order to prove or disprove this assumption, we instantiate random ideal lattices that allow to test algorithms that solve SVP and its approximate version. The Ideal Lattice Challenge allows online submission of short vectors to enter a hall of fame for full comparison. We adjoin a set of first experiments and a first comparison of ideal and regular lattices.
Abstract. Lattice-based cryptography is one of the candidates in the area of post-quantum cryptography. Cryptographic schemes with security reductions to hard lattice problems (like the Shortest Vector Problem SVP) offer an alternative to recent number theory-based schemes. In order to guarantee asymptotic efficiency, most lattice-based schemes are instantiated using polynomial rings over integers. These lattices are called ideal lattices. It is assumed that the hardness of lattice problems in lattices over integer rings remains the same as in regular lattices. In order to prove or disprove this assumption, we instantiate random ideal lattices that allow to test algorithms that solve SVP and its approximate version. The Ideal Lattice Challenge allows online submission of short vectors to enter a hall of fame for full comparison. We adjoin a set of first experiments and a first comparison of ideal and regular lattices.
The security of cryptographic protocols is based on the presumed difficulty of algorithmic problems. Among those identified so far, some of the best problems to serve as a foundation for quantum-proof cryptography come from lattices. Lattices are a mathematical structure defined as a set of space vectors generated by integer combinations of a finite number of linearly independent real vectors (its basis). A typical example of a related security problem is the Shortest Vector Problem (SVP). Given a base of an n-dimensional lattice, find a non-zero short vector. For efficiency reasons, these problems are restricted to lattices arising from number theory, known as structured lattices. As the security assumptions for these particular lattices are different from those for unstructured lattices, it is necessary to study them specifically, which is the aim of this thesis.We have studied the case of NTRU and uSVP modules in rank 2, proving that the SVP problem is equivalent on these two families of lattices. We also show a worst-case to average-case reduction for rank-2 uSVP lattices. Then we show that solving SVP on a prime ideal of small norm is no easier than solving SVP on any ideal.
The imminent threat of quantum computing poses a significant challenge to the security of current public-key cryptographic systems, most notably RSA. These classical algorithms rely on mathematical problems, such as integer factorization and discrete logarithms, which are efficiently solvable by Shor's algorithm on a sufficiently powerful quantum computer. This paper explores ideal lattices as a robust post-quantum foundation for asymmetric cryptography, offering a compelling alternative to traditional schemes. We delve into the mathematical underpinnings of ideal lattices, including the Ring Learning With Errors (RLWE) and Shortest Vector Problem (SVP), which are conjectured to be hard even for quantum computers. The advantages of ideal lattice-based cryptography, such as its quantum resistance, efficiency, and versatility in constructing various cryptographic primitives, are thoroughly discussed. We examine prominent examples like NTRU and CRYSTALS-Kyber, highlighting their design principles and performance characteristics. Furthermore, the paper addresses the ongoing standardization efforts by NIST and the practical considerations for migrating to lattice-based schemes. The objective is to present a comprehensive analysis demonstrating why ideal lattices are a leading candidate for securing digital communications in the quantum era, providing a secure and efficient pathway beyond the limitations of RSA.
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.
The lattice-based encryption paradigm (LBE) represents a new security model emphasizing flexible development, high levels of security, and efficient computations. It is now becoming more commonly seen as an alternative to other systems of post-quantum cryptography due to its foundations for both security (i.e., there is a high degree of confidence that lattice-based cryptography is secure) and efficiency with respect to computation. The theoretical basis for this report is related to the difficulty of solving the short vector problem (SVP) and other cryptographic hard problems concerning lattices. The report focuses on the hardness of lattice-based cryptography relative to three sub-components of the theoretical work: the relationship between geometric number theory and lattice-based systems, the application of AI to help determine parameters and the efficiencies of finding new lattice reduction algorithms, and finally estimating the difficulty to solve lattice problems. This report reviews the inherent difficulty of short vector problems (SVPs) and other kinds of SVPs, such as the closest vector problem. The report analyzes how to reduce both the time and space complexities for classical and quantum computing solutions through the discovery of new techniques based upon number-theoretic concepts. The report also discusses a number of examples, including digital signature products, encryption systems and homomorphic encryption systems as primitive constructions of cryptography based on lattices or through the properties of the lattice structures. Finally, based on number-theoretical relationships among complex numbers, the report provides a new theoretical approach to the construction of lattice-based systems and their strength relative to other types of cryptographic systems via the provision of theoretical bounds related to the lengths of lattice basis vectors.As a result of this research into bridging the divide between theoretical and practical cryptography, the results show that lattice-based systems have the capability of providing the strength to withstand quantum attacks. Through this research study, there is an improvement in how we understand mathematically the difficulty of lattices and how to create post-quantum secure cryptographic techniques through the use of lattices...
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.
In this thesis we will discuss hard computational problems in lattice theory and relate them to cryptographic constructions. Many of these problems enjoy average-case hardness which makes them attractive for cryptography. In addition, lattice based cryptography is a candidate for post-quantum cryptography, as there is no known quantum algorithm which breaks various hardness theorems.\n\nWe build a foundation in algebraic number theory to have the required background to discuss schemes based on discrete algebraic structures. These structures are free Z-modules which permits unique factorization in prime ideals. We relate this algebraic number theory to lattices in R^n so we can use the theory from algebra to our advantage.\n\nWe then define some standard hard computational lattice problems and show how many of these are related to each other. We prove that these problems are at least as hard as finding the shortest vector of a lattice, which we conjecture is computationally infeasible. We then prove a quantum reduction the learning with errors problem, a problem in machine learning . We also show that there is a similar reduction for a variant of this problem over more general rings.