Ideal Lattices Research Articles

Progress and Applications of Fully Homomorphic Encryption

In the era of digital transformation, marked by the widespread adoption of cloud-based technologies such as cloud computing and storage, ensuring data security and protecting user privacy have emerged as critical industry concerns. Fully Homomorphic Encryption (FHE) addresses these issues by allowing various computations to be performed on ciphertexts without the need for decryption, thus safeguarding data security and privacy. This paper explores the mathematical and algorithmic foundations essential for understanding FHE. It provides a detailed analysis of the advancements in FHE schemes over the past decade, focusing on various mathematical problems, including ideal lattices and integers. The discussion extends to three practical applications: cloud computing, machine learning, and electronic voting, highlighting the progress and exploring potential future research avenues and developmental strategies in the field of FHE. This comprehensive review not only underscores the significance of FHE in enhancing data security but also charts a path for its future exploration and integration into emerging technologies.

Fracture mechanics of bi-material lattice metamaterials

The advent of additive manufacturing technology empowers precise control of multi-material components or specific defects in lightweight lattice metamaterials, however, fracture mechanics and toughening design strategies in such metamaterials remain enigmatic. By incorporating theoretical analysis, numerical simulation, and experimental investigation, our study reveals that stretch-bend synergistic strut deformations caused by bi-material components or topology defects contribute notably tougher lattice structures surpassing its ideal single-material lattices. A peak fracture energy at a critical modulus ratio was found in a designed bi-material lattice composed of triangular soft struts and hexagonal stiff struts, which originates from the shift of fracture modes at crack tip from strut bending to stretching dominated failure modes as the modulus of soft struts increases, where the compromise in competition between bending-enhanced and stretching-weakened energy dissipations of struts deformations results in the maximized fracture energy. A parametric design protocol was proposed to optimize fracture energy of bi-material lattices through tuning the modulus ratio and relative density. Furthermore, the concept of stretch-bend synergistic toughening can also be applied to make tougher single-material lattices with specific topological defects. Our findings not only provide physical insights into directing crack propagation but also provide quantitative guidance to optimize fracture resistance within low-density tough lattice metamaterials.

An Exploration of the Phonon Frequency Spectrum and Born–von Karman Periodic Boundary Conditions in 1D and 2D Lattice Systems Using a Computational Approach

Periodic boundary conditions (PBCs) are a pivotal concept in the treatment of ideal lattices of infinite extent as a finite lattice. Most undergraduate texts that delve into the analytical treatment of lattice dynamics solve the problem in the reciprocal space with an implicit assumption of PBCs. While the traditional approach centered in the reciprocal or [Formula: see text]-space is elegant for simple one-dimensional lattice structures, the method becomes mathematically cumbersome for more complex lattices with real-world applications, such as a honeycomb lattice involving a [Formula: see text] secular determinant. In this work, we have explored the solution of lattice dynamics in direct space numerically, by constructing unit cells through an explicit implementation of Born–von Karman PBCs and solving the lattice dynamical equations in the displacement–time domain using the fourth-order Runge–Kutta algorithm. The Fast Fourier Transform technique is then used to obtain the phonon frequency spectrum corresponding to the computed instantaneous displacements. The fidelity of the model is validated by the agreement of the computational results obtained for the phonon spectrum at high symmetry points in the Brillouin zone with the analytical values for the given problem. Initially, the dynamics and the phonon spectrum are explored for monatomic and diatomic linear lattices which serve to act as introductory cases for the approach before investigating the 2D monatomic square and honeycomb lattices using the nearest and next-nearest neighbor approximations. A short-range force constant model employing the central forces and angular forces is used for the monatomic square lattice to account for the interatomic interactions. Our computational approach is expected to be physically more intuitive for a student for visualizing the periodicity of a given lattice and understanding its related dynamics. The approach demonstrates the usage of Born–von Karman PBCs in constructing a unit cell that effectively captures the dynamics of a lattice system, complementing the traditional secular determinant formalism. The work also illustrates an instance of how the computer programming can be used as a tool to empower the students to explore abstract concepts in physics. A honing of programming and computational skills of students enables them to interact with dynamic models in physics and gain deeper insight into the real-world applications of physics principles. Target group of this work is the students and educators at undergraduate level who seek a thorough and didactic comprehension of lattice dynamics.

Ciphertext policy attribute-based encryption scheme supporting Boolean circuits over ideal lattices

Considering the existence of fast implementation methods for multiplication operations over ideal lattices, we constructed a selectively secure ciphertext policy attribute-based encryption scheme supporting Boolean circuits based non-monotonic linear secret sharing scheme. It uses the trapdoor generation algorithm TrapGen to generate the public parameters and the preimage sampling algorithm SamplePre to embed the public parameters and randomness into the user’s secret key, which achieves the randomization of the secret key. The sharing and reconstruction of the secret in the encryption and decryption algorithm are achieved by a non-monotonic linear secret sharing scheme. Compared to the existing ciphertext policy attribute-based encryption schemes based on a similar sampling algorithm, the size of the ciphertext is significantly reduced.

Ideal lattices in semigroup of doubly stochastic matrices

In this paper, we illustrate the rule for finding number of idempotents in the doubly stochastic matrix semigroup [Formula: see text] and also locate the idempotents in the case of [Formula: see text] and [Formula: see text]. Further, we describe idempotent generated ideals of these semigroups and it is shown that the idempotent generated ideals form a lattice.

Shorter ZK-SNARKs from square span programs over ideal lattices

Zero-knowledge succinct non-interactive arguments of knowledge (zk-SNARKs) are cryptographic protocols that offer efficient and privacy-preserving means of verifying NP language relations and have drawn considerable attention for their appealing applications, e.g., verifiable computation and anonymous payment protocol. Compared with the pre-quantum case, the practicability of this primitive in the post-quantum setting is still unsatisfactory, especially for the space complexity. To tackle this issue, this work seeks to enhance the efficiency and compactness of lattice-based zk-SNARKs, including proof length and common reference string (CRS) length. In this paper, we develop the framework of square span program-based SNARKs and design new zk-SNARKs over cyclotomic rings. Compared with previous works, our construction is without parallel repetition and achieves shorter proof and CRS lengths than previous lattice-based zk-SNARK schemes. Particularly, the proof length of our scheme is around 23.3%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$23.3\\%$$\\end{document} smaller than the recent shortest lattice-based zk-SNARKs by Ishai et al. (in: Proceedings of the 2021 ACM SIGSAC conference on computer and communications security, pp 212–234, 2021), and the CRS length is 3.6×\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$3.6\ imes$$\\end{document} smaller. Our constructions follow the framework of Gennaro et al. (in: Proceedings of the 2018 ACM SIGSAC conference on computer and communications security, pp 556–573, 2018), and adapt it to the ring setting by slightly modifying the knowledge assumptions. We develop concretely small constructions by using module-switching and key-switching procedures in a novel way.

Offline/online attribute-based searchable encryption scheme from ideal lattices for IoT

Offline/online attribute-based searchable encryption scheme from ideal lattices for IoT

Concrete analysis of approximate ideal-SIVP to decision ring-LWE reduction

<p style='text-indent:20px;'>A seminal 2013 paper by Lyubashevsky, Peikert, and Regev proposed basing post-quantum cryptography on ideal lattices and supported this proposal by giving a polynomial-time security reduction from the approximate Shortest Independent Vectors Problem (SIVP) to the Decision Learning With Errors (DLWE) problem in ideal lattices. We give a concrete analysis of this multi-step reduction. We find that the tightness gap in the reduction is so great as to vitiate any meaningful security guarantee, and we find reasons to doubt the feasibility in the foreseeable future of the quantum part of the reduction. In addition, when we make the reduction concrete it appears that the approximation factor in the SIVP problem is far larger than expected, a circumstance that causes the corresponding approximate-SIVP problem most likely not to be hard for proposed cryptosystem parameters. We also discuss implications for systems such as Kyber and SABER that are based on module-DLWE.</p>

A Robust and Lightweight Privacy-Preserving Data Aggregation Scheme for Smart Grid

Privacy-preserving data aggregation (PPDA) enables data availability and privacy preservation simultaneously in smart grid. However, existing methods, such as masking and homomorphic encryption, cannot simultaneously offer strong privacy preservation, fault tolerance for both smart meters and aggregators, verifiable aggregation, and lightweight encryption. To tackle these challenges, we design HTV-PRE, a homomorphic threshold proxy re-encryption scheme with re-encryption verifiability. HTV-PRE involves only linear operations and resists quantum attacks after being instanced by ideal lattices. By leveraging HTV-PRE, we propose a robust and lightweight data aggregation scheme with strong privacy preservation for smart grid. Robustness ensures fault tolerance and error detection. Even if some smart meters or aggregators are faulty, data aggregation can still work without imposing expensive computation on other smart meters or requiring additional trust assumptions. Additionally, to detect aggregators' errors, a proof for the aggregated result is presented so that anyone can verify whether the result has been correctly computed or not. The verifiable aggregation adds no computation/communication overhead on the user side. The performance evaluations demonstrate that our PPDA scheme significantly offloads computation overhead from smart meters and control center to the edge, and its user encryption is up to 4x faster than existing approaches.

Limits of Ohm’s law in ultra-pure metals with almost ideal crystal lattices

By electric transport measurements on high-purity bulk molybdenum single crystals at helium-temperature, it is shown that when approaching the ideal metallic crystal structure, there are deviations from Ohm’s law that can exceed 30% even at low current densities ≅3A/mm2. This is due to the intrinsic circular magnetic field of the measuring current in the range of B≅1mT, which generates a magnetoresistance according to the two-band conduction model. With increasing lattice vibrations and/or crystal defects, this inherent uncertainty finally reduces to immeasurably small values. Finally, Ohm’s law for a nearly ideal lattice structure can be strictly exact only in the theoretical limiting case of an infinite lattice plane, because then the magnetic fields of all electrons of the current compensate to zero.

Well-Rounded ideal lattices of cyclic cubic and quartic fields

In this paper, we find criteria for when cyclic cubic and cyclic quartic fields have well-rounded ideal lattices. We show that every cyclic cubic field has at least one well-rounded ideal. We also prove that there exist families of cyclic quartic fields which have well-rounded ideals and explicitly construct their minimal bases. In addition, for a given prime number $p$, if a cyclic quartic field has a unique prime ideal above $p$, then we provide the necessary and sufficient conditions for that ideal to be well-rounded. Moreover, in cyclic quartic fields, we provide the prime decomposition of all odd prime numbers and construct an explicit integral basis for every prime ideal.

Topology optimization of fibre reinforced polymer lattice structures for additive manufacturing

The material anisotropy induced by the related additive manufacturing (AM) extrusion process and the geometric errors introduced into the unit lattices by the stair-stepping artifacts were coupled in the topology optimization model to ensure the accuracy and reliability of the optimal design for the fibre reinforced polymer lattice structures. The material properties of the 3D printed carbon fibre reinforced polymers (CFRP) were firstly systematically investigated by extensive tensile tests aided by the digital image correlation (DIC) system, through which the microscale material anisotropy was clarified. Moreover, the accurate orthotropic material models were introduced into the as-fabricated mesoscale unit lattices with geometric errors to obtain the effective mechanical properties. Finally, mechanical performance of the macroscale structures was optimized by the integration of the ideal and as-fabricated unit lattices with the homogenization-based topology optimization method. The AM process-induced material anisotropy and geometric errors result in significant deviations of the mechanical property of the topologically optimized macrostructures and more accurate topology optimization results can be achieved considering the microscale material anisotropy and the geometric error in the mesoscale unit lattice. The results obtained in this investigation indicate that it is essential to comprehensively consider the AM process-induced microscale material anisotropy and mesoscale geometric errors in the topology optimization of fibre reinforced polymer lattice structures to improve the accuracy and reliability of the optimal design.

Quantum-safe three-party lattice based authenticated key agreement protocol for mobile devices

Mobile-based web apps have gained popularity among phone devices because of the rapid advancement of mobile communication and equipment capabilities. Users may utilize their handsets to obtain different types of online services through the Internet at any time and from anywhere. The last three decades have seen widespread use of the three-party AKA protocol, built on asymmetric-key cryptography to produce a private communication channel via any cellular connection between two mobile users. The server can help two mobile users with the assistance of a three-party authentication protocol so that they may create a high-entropy shared private session key. According to several recent studies, all the conventional public-key cryptosystem-based three-party AKA schemes could not withstand attacks due to quantum computing. To get resistance to quantum attacks, Islam and Basu (2021) proposed a new password-based three-party authentication protocol for mobile users using ideal lattices. We improve the efficiency of lattice-based authenticated key exchange protocol and the security of this protocol by making it resistant to password-guessing attacks. A comparative study along with performance and security evaluation is also presented, which demonstrates the applicability of the proposed design.

Lie Structures and Chain Ideal Lattices

The purpose of this paper is twofold. Firstly, to emphasise that the class of Lie algebras with chain lattices of ideals are elementary blocks in the embedding or decomposition of Lie algebras with finite lattice of ideals. Secondly, to show that the number of Lie algebras of this class is large and they support other types of Lie structures. Beginning with general examples and algebraic decompositions, we focus on computational algorithms to build Lie algebras in which the lattice of ideals is a chain. The chain condition forces gradings on the nilradicals of this class of algebras. Our algorithms yield to several positive naturally graded parametric families of Lie algebras. Further generalizations and other kind of structures will also be discussed.

On the security of ring learning with error‐based key exchange protocol against signal leakage attack

AbstractDue to the rapid development of mobile communication and hardware technologies, several mobile‐based web applications have gained popularity among mobile users. Mobile users can utilize these devices to access numerous services over the Internet. To ensure secure communication, different key exchange and authentication (KEA) protocols are proposed and frequently used. However, due to the advent of quantum computers, numerous quantum‐safe KEA protocols are also developed using various complex mathematical problems in ideal lattices. As it is an emerging and developing area, we analyze the security of recently suggested ring learning with error based KEA protocols. The goal of this study is to gain a comprehensive understanding of quantum‐safe KEA mechanisms. For our study, we have considered Dharminder's LWE‐based KEA mechanism and Dharminder and Chandran's LWE‐based KEA mechanism. These protocols enable effective communication and provide a better means for safely transmitting messages between user and server. However, we have discovered that a fundamental security weakness in these methods makes them vulnerable to signal leakage attacks (SLA). Based on our analysis, we demonstrated security weakness against SLA and provide the road‐map for secure construction.

A Novel Linkable Ring Signature on Ideal Lattices

In this paper, a novel linkable ring signature scheme is constructed. The hash value of the public key in the ring and the signer's private key are based on random numbers. This setting makes it unnecessary to set the linkable label separately for our constructed scheme. When judging the linkability, it is necessary to determine whether the number of the intersections of the two sets reaches the threshold related to the number of the ring members. In addition, under the random oracle model, the unforgeability is reduced to the SVPγ problem. The anonymity is proved based on the definition of statistical distance and its properties.

An Unbounded Fully Homomorphic Encryption Scheme Based on Ideal Lattices and Chinese Remainder Theorem

An Unbounded Fully Homomorphic Encryption Scheme Based on Ideal Lattices and Chinese Remainder Theorem

Dynamic group signature scheme using ideal lattices

Dynamic group signature scheme using ideal lattices

Characterization of Nested Lattice Codes Through Quotient Groups Relative to Normal Fuzzy Subgroups for Channel Quantization

Interference is in general, an undesirable obstacle to communication in wireless communication networks. Therefore, in order to realize interference alignment, this work intends to characterize classical codes through the framework of fuzzy set theory for quantizing real-valued channel coefficients. Thenceforward, from the classical framework nested real lattice codes are constructed by using construction A of nested real ideal lattices and, therefore, through normal fuzzy subgroups this work shows that the constructed quotient groups of the nested real ideal lattices relative to normal fuzzy subgroups of these nested real ideal lattices are nested vector spaces and, consequently, such nested quotient groups have the structure of a linear code, are isomorphic to the nested real lattice codes and defined by linear fuzzy codes. Through such an isomorphism and the construction of the nested quotient groups relative to normal fuzzy subgroups, it is possible to conclude that the codewords of these nested linear fuzzy codes are in a one-to-one correspondence to membership vectors. Through the constructed nested quotient groups, that is, nested linear fuzzy codes, this work supplies how to quantize real-valued channel coefficients onto these nested quotient groups in order to realize interference alignment and, consequently, it is possible to encode an arbitrary real-valued channel coefficient by a codeword from a linear fuzzy code.

Scalable CCA-secure public-key authenticated encryption with keyword search from ideal lattices in cloud computing

Public-key authenticated encryption with keyword search (PAEKS) is an important cryptographic primitive for realizing effective and confidential searches on encrypted data in cloud computing. Several PAEKS schemes with various appealing functionalities have been proposed in the literature. Nevertheless, most of them are based on the classical number-theoretic assumptions and are impossible to resist quantum attacks. Their security is only proven against chosen-keyword attacks and is considered insufficient. Fortunately, ring-LWE and ring-ISIS problems reduced from the worst-case ideal lattices are believed to be post-quantum secure and could be improved computational efficiency in constructions. In this paper, we propose a scalable PAEKS scheme based on ring-LWE and ring-ISIS by adopting the ring analog of lattice algorithms of both digital signature and searchable encryption schemes, in which signature algorithms are used to authenticate ciphertext, making our PAEKS resistant to insider keyword guessing attacks. The scheme is also proved secure against chosen-ciphertext attacks and token privacy attacks. In addition, we extend the PAEKS scheme to conjunctive keywords and multi-user settings to allow for realistic promotion. We finally implement our schemes and compare them with related counterparts, which shows that our constructions are efficient in practical applications.