Inspired by quantum annealing, digital annealing computers specified for annealing computations have been realized on a large scale, such as the Digital Annealer (DA) developed by Fujitsu and the CMOS Annealing Machine developed by Hitachi. With the progress achieved using these computers, it has become necessary to estimate the computational hardness of cryptographic problems. This paper focuses on lattice problems, such as the closest vector problem (CVP) and shortest vector problem (SVP), which are a class of optimization problems. These problems form the basis of the security of lattice-based cryptography, which is a prime candidate for the NIST post-quantum cryptography standardization. For these lattice problems, we propose methods for generating an Ising model and solving the Ising model on annealing computers with a bit representation as the input, which represents encodings to map each integer variable in the SVP into binary variables. We propose two methods for SVPs, a basic method and a variant incorporating approximately the concept of the classical lattice enumeration. In our experimental results obtained using the second-generation DA, we succeeded in finding a shortest nonzero lattice vector in 40- and 45-dimensional lattices in the Darmstadt SVP Challenge. The basic method with a hybrid bit representation was the fastest among our methods with a bit representation, and the expected running time was estimated as 664 and 13,750 seconds for the 40- and 45-dimensional lattices, respectively. These results provide a benchmark for solving the SVP with annealing computers.
Inspired by quantum annealing, digital annealing computers specified for annealing computations have been realized on a large scale, such as the Digital Annealer (DA) developed by Fujitsu and the CMOS Annealing Machine developed by Hitachi. With the progress achieved using these computers, it has become necessary to estimate the computational hardness of cryptographic problems. This paper focuses on lattice problems, such as the closest vector problem (CVP) and shortest vector problem (SVP), which are a class of optimization problems. These problems form the basis of the security of lattice-based cryptography, which is a prime candidate for the NIST post-quantum cryptography standardization. For these lattice problems, we propose methods for generating an Ising model and solving the Ising model on annealing computers with a bit representation as the input, which represents encodings to map each integer variable in the SVP into binary variables. We propose two methods for SVPs, a basic method and a variant incorporating approximately the concept of the classical lattice enumeration. In our experimental results obtained using the second-generation DA, we succeeded in finding a shortest nonzero lattice vector in 40- and 45-dimensional lattices in the Darmstadt SVP Challenge. The basic method with a hybrid bit representation was the fastest among our methods with a bit representation, and the expected running time was estimated as 664 and 13,750 seconds for the 40- and 45-dimensional lattices, respectively. These results provide a benchmark for solving the SVP with annealing computers.
In 1994, P. Shor discovered quantum algorithms that can break both the RSA cryptosystem and the ElGamal cryptosystem. In 2007, D-Wave demonstrated the first quantum computer. These events and further developments have brought a crisis to secret communication. In 2016, the National Institute of Standards and Technology (NIST) launched a global project to solicit and select a handful of encryption algorithms with the ability to resist quantum computer attacks. In 2022, it announced four candidates, CRYSTALS-Kyber, CRYSTALS-Dilithium, Falcon, and Sphincs+, for post-quantum cryptography standards. The first three are based on lattice theory and the last on a hash function. The security of lattice-based cryptosystems relies on the computational complexity of the shortest vector problem (SVP), the closest vector problem (CVP), and their generalizations. As we will explain, the SVP is a ball-packing problem, and the CVP is a ball-covering problem. Furthermore, both the SVP and CVP are equivalent to arithmetic problems for positive definite quadratic forms. This paper will briefly describe the mathematical problems on which lattice-based cryptography is built so that cryptographers can extend their views and learn something useful.
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...
Abstract The shortest vector problem (SVP) is one of the lattice problems and is the mathematical basis for lattice-based cryptography, which is expected to be post-quantum cryptography. The SVP can be mapped onto the Ising problem, which in principle can be solved by quantum annealing (QA). However, one issue in solving the SVP using QA is that the solution of the SVP corresponds to the first excited state of the problem Hamiltonian. Therefore, QA, which searches for ground states, cannot provide a solution with high probability. In this paper, we propose to adopt an excited-state search of the QA to solve the shortest vector problem. We numerically show that the excited-state search provides a solution with a higher probability than the ground-state search.
Lattice-based cryptography has emerged as one of the most prominent candidates for postquantum cryptography, projected to be secure against the imminent threat of large-scale fault-tolerant quantum computers. The Shortest Vector Problem (SVP) is to find the shortest nonzero vector in a given lattice. It is fundamental to lattice-based cryptography and believed to be hard even for quantum computers. We study a natural generalization of the SVP known as the K -Densest Sublattice Problem ( K -DSP): to find the densest K -dimensional sublattice of a given lattice. We formulate K -DSP as finding the first excited state of a Z -basis Hamiltonian, making K -DSP amenable to investigation via an array of quantum algorithms, including Grover search, quantum Gibbs sampling, adiabatic, and variational quantum algorithms. The complexity of the algorithms depends on the basis through which the input lattice is presented. We present a classical polynomial-time algorithm that takes an arbitrary input basis and preprocesses it into inputs suited to quantum algorithms. With preprocessing, we prove that O ( K N 2 ) qubits suffice for solving K -DSP for N -dimensional input lattices. We empirically demonstrate the performance of a quantum approximate optimization algorithm K -DSP solver for low dimensions, highlighting the influence of a good preprocessed input basis. We then discuss the hardness of K -DSP in relation to the SVP, to see if there is a reason to build postquantum cryptography on K -DSP. We devise a quantum algorithm that solves K -DSP with runtime exponent ( 5 K N log 2 N ) / 2 . Therefore, for fixed K , K -DSP is no more than polynomially harder than the SVP. The central insight we use is similar in spirit to those underlying the best-known classical K -DSP algorithm due to Dadush and Micciancio. Whether the exponential dependence on K can be lowered remains an open question. Published by the American Physical Society 2024
The advent of quantum computing poses a significant existential threat to the security foundations of current public-key cryptography, most notably the widely adopted RSA algorithm. Shor's algorithm, specifically, can efficiently break the mathematical problems underlying RSA and elliptic curve cryptography, necessitating a paradigm shift towards quantum-resistant solutions. This paper explores lattice-based cryptography as the leading candidate for this post-quantum era. We delve into the fundamental mathematical principles of lattices, including hard problems like the Shortest Vector Problem (SVP), Closest Vector Problem (CVP), Short Integer Solution (SIS), and Learning With Errors (LWE), which form the bedrock of their security. The paper reviews prominent lattice-based schemes such as NTRU, CRYSTALS-Kyber, CRYSTALS-Dilithium, and Falcon, highlighting their construction, security properties, and performance characteristics. We analyze their resilience against both classical and quantum attacks, discuss their progress in the NIST Post-Quantum Cryptography Standardization process, and evaluate their practical implications and implementation challenges. The discussion emphasizes the provable security, versatility, and efficiency of lattice-based methods, positioning them as the robust and essential successor to RSA for safeguarding digital communications and data in a quantum-enabled future.
A fundamental computational problem is to find a shortest non-zero vector in Euclidean lattices, a problem known as the Shortest Vector Problem (SVP). This problem is believed to be hard even on quantum computers and thus plays a pivotal role in post-quantum cryptography. In this work we explore how (efficiently) Noisy Intermediate Scale Quantum (NISQ) devices may be used to solve SVP. Specifically, we map the problem to that of finding the ground state of a suitable Hamiltonian. In particular, (i) we establish new bounds for lattice enumeration, this allows us to obtain new bounds (resp. estimates) for the number of qubits required per dimension for any lattices (resp. random q-ary lattices) to solve SVP; (ii) we exclude the zero vector from the optimization space by proposing (a) a different classical optimisation loop or alternatively (b) a new mapping to the Hamiltonian. These improvements allow us to solve SVP in dimension up to 28 in a quantum emulation, significantly more than what was previously achieved, even for special cases. Finally, we extrapolate the size of NISQ devices that is required to be able to solve instances of lattices that are hard even for the best classical algorithms and find that with approximately 103 noisy qubits such instances can be tackled.