Lattice Algorithm Research Articles

A semi-numerical algorithm for the homology lattice and periods of complex elliptic surfaces over [formula omitted]

We provide an algorithm for computing a basis of homology of elliptic surfaces over PC1 that is sufficiently explicit for integration of periods to be carried out. This allows the heuristic recovery of several algebraic invariants of the surface, notably the Néron–Severi lattice, the transcendental lattice, the Mordell–Weil group and the Mordell–Weil lattice. This algorithm comes with a SageMath implementation.

Ideas of lattice-basis reduction theory for error-stable Bravais lattice determination and abinitio indexing.

In ab initio indexing, for a given diffraction/scattering pattern, the unit-cell parameters and the Miller indices assigned to reflections in the pattern are determined simultaneously. `Ab initio' means a process performed without any good prior information on the crystal lattice. Newly developed ab initio indexing software is frequently reported in crystallography. However, it is not widely recognized that use of a Bravais lattice determination method, which is tolerant of experimental errors, can simplify indexing algorithms and increase their success rates. One of the goals of this article is to collect information on the lattice-basis reduction theory and its applications. The main result is a Bravais lattice determination algorithm for 2D lattices, along with a mathematical proof that it works even for parameters containing large observational errors. It uses two lattice-basis reduction methods that seem to be optimal for different symmetries, similarly to the algorithm for 3D lattices implemented in the CONOGRAPH software. In indexing, a method for error-stable unit-cell identification is also required to exclude duplicate solutions. Several methods are introduced to measure the difference in unit cells known in crystallography and mathematics.

Quantum simulation of dissipation for Maxwell equations in dispersive media

The dissipative character of an electromagnetic medium breaks the unitary evolution structure that is present in lossless, dispersive optical media. In dispersive media, dissipation appears in the Schrödinger representation of classical Maxwell equations as a sparse diagonal operator occupying an r-dimensional subspace. A first order Suzuki-Trotter approximation for the evolution operator enables us to isolate the non-unitary operators (associated with dissipation) from the unitary operators (associated with lossless media). The unitary operators can be implemented through qubit lattice algorithm (QLA) on n qubits, based on the discretization and the dimensionality of the pertinent fields. However, the non-unitary-dissipative part poses a challenge both physically and computationally on how it should be implemented on a quantum computer. In this paper, two probabilistic dilation algorithms are considered for handling the dissipative operators. The first algorithm is based on treating the classical dissipation as a linear amplitude damping-type completely positive trace preserving (CPTP) quantum channel where an unspecified environment interacts with the system of interest and produces the non-unitary evolution. Therefore, the combined system-environment is now closed, and must undergo unitary evolution in the dilated space. The unspecified environment can be modeled by just one ancillary qubit, resulting in an implementation scaling of O(2n−1n2) elementary gates for the total system-environment unitary evolution operator. The second algorithm approximates the non-unitary operators by the Linear Combination of Unitaries (LCU). On exploiting the diagonal structure of the dissipation, we obtain an optimized representation of the non-unitary part, which requires O(2n) elementary gates. Applying the LCU method for a simple dielectric medium with homogeneous dissipation rate, the implementation scaling can be further reduced into O[poly(n)] basic gates. For the particular case of weak dissipation we show that our proposed post-selective dilation algorithms can efficiently delve into the transient evolution dynamics of dissipative systems by calculating the respective implementation circuit depth. A connection of our results with the non-linear-in-normalization-only (NINO) quantum channels is also presented.

From approximate to exact integer programming

AbstractApproximate integer programming is the following: For a given convex body $$K \subseteq {\mathbb {R}}^n$$ K ⊆ R n , either determine whether $$K \cap {\mathbb {Z}}^n$$ K ∩ Z n is empty, or find an integer point in the convex body $$2\cdot (K - c) +c$$ 2 · ( K - c ) + c which is K, scaled by 2 from its center of gravity c. Approximate integer programming can be solved in time $$2^{O(n)}$$ 2 O ( n ) while the fastest known methods for exact integer programming run in time $$2^{O(n)} \cdot n^n$$ 2 O ( n ) · n n . So far, there are no efficient methods for integer programming known that are based on approximate integer programming. Our main contribution are two such methods, each yielding novel complexity results. First, we show that an integer point $$x^* \in (K \cap {\mathbb {Z}}^n)$$ x ∗ ∈ ( K ∩ Z n ) can be found in time $$2^{O(n)}$$ 2 O ( n ) , provided that the remainders of each component $$x_i^* \mod \ell $$ x i ∗ mod ℓ for some arbitrarily fixed $$\ell \ge 5(n+1)$$ ℓ ≥ 5 ( n + 1 ) of $$x^*$$ x ∗ are given. The algorithm is based on a cutting-plane technique, iteratively halving the volume of the feasible set. The cutting planes are determined via approximate integer programming. Enumeration of the possible remainders gives a $$2^{O(n)}n^n$$ 2 O ( n ) n n algorithm for general integer programming. This matches the current best bound of an algorithm by Dadush (Integer programming, lattice algorithms, and deterministic, vol. Estimation. Georgia Institute of Technology, Atlanta, 2012) that is considerably more involved. Our algorithm also relies on a new asymmetric approximate Carathéodory theorem that might be of interest on its own. Our second method concerns integer programming problems in equation-standard form $$Ax = b, 0 \le x \le u, \, x \in {\mathbb {Z}}^n$$ A x = b , 0 ≤ x ≤ u , x ∈ Z n . Such a problem can be reduced to the solution of $$\prod _i O(\log u_i +1)$$ ∏ i O ( log u i + 1 ) approximate integer programming problems. This implies, for example that knapsack or subset-sum problems with polynomial variable range$$0 \le x_i \le p(n)$$ 0 ≤ x i ≤ p ( n ) can be solved in time $$(\log n)^{O(n)}$$ ( log n ) O ( n ) . For these problems, the best running time so far was $$n^n \cdot 2^{O(n)}$$ n n · 2 O ( n ) .

Enhancing LOCA Breach Size Diagnosis with Fundamental Deep Learning Models and Optimized Dataset Construction

In nuclear power plants, the loss-of-coolant accident (LOCA) stands out as the most prevalent and consequential incident. Accurate breach size diagnosis is crucial for the mitigation of LOCAs, and identifying the cause of an accident can prevent catastrophic consequences. Traditional methods mostly focus on combining model algorithms and utilize intricate composite model neural network architectures. However, it is crucial to investigate whether greater complexity necessarily leads to better performance. In addition, the consideration of the impact of dataset construction and data preprocessing on model performance is also needed for model building. This paper proposes a framework named DeepLOCA-Lattice to experiment with different preprocessing approaches to fundamental deep learning models for a comprehensive analysis of the diagnosis of LOCA breach size. The DeepLOCA-Lattice involves data preprocessing via the lattice algorithm and equal-interval partitioning and deep-learning-based models, including the multi-layer perceptron (MLP), recurrent neural networks (RNNs), convolutional neural networks (CNNs), and the transformer model in LOCA breach size diagnosis. After conducting rigorous ablation experiments, we have discovered that even rudimentary foundational models can achieve accuracy rates that exceed 90%. This is a significant improvement when compared to the previous models, which yield an accuracy rate of lower than 50%. The results interestingly demonstrate the superior performance and efficacy of the fundamental deep learning model, with an effective dataset construction approach. It elucidates the presence of a complex interplay among diagnostic scales, sliding window size, and sliding stride. Furthermore, our investigation reveals that the model attains its highest accuracy within the discussed range when utilizing a smaller sliding stride size and a longer sliding window length. This study could furnish valuable insights for constructing models for LOCA breach size estimation.

AIOL: An Improved Orthogonal Lattice Algorithm for the General Approximate Common Divisor Problem

The security of several fully homomorphic encryption (FHE) schemes depends on the intractability assumption of the approximate common divisor (ACD) problem over integers. Subsequent efforts to solve the ACD problem as well as its variants were also developed during the past decade. In this paper, an improved orthogonal lattice (OL)-based algorithm, AIOL, is proposed to solve the general approximate common divisor (GACD) problem. The conditions for ensuring the feasibility of AIOL are also presented. Compared to the Ding–Tao OL algorithm, the well-known LLL reduction method is used only once in AIOL, and when the error vector r is recovered in AIOL, the possible difference between the restored and the true value of p is given. Experimental comparisons between the Ding-Tao algorithm and ours are also provided to validate our improvements.

Vertical height growth mechanism of hydraulic fractures in laminated shale oil reservoirs based on 3D discrete lattice modeling

Bedding planes are abundant in shale oil reservoirs, but the intrinsic mechanism of fracture-height containment by these weak interfaces remains unclear. To investigate the effects of interface properties, stress conditions, and fracturing fluid viscosity on the vertical propagation of fracture heights in laminated shale oil reservoirs, a three-dimensional hydro-mechanical coupling numerical model was developed. The model is based on the 3D discrete lattice algorithm (DLA), which replaces the balls and contacts in the conventional synthetic rock mass model (SRM) with a lattice consisting of spring-connected nodes, resulting in improved computational efficiency. Additionally, the interaction between hydraulic fractures and bedding planes is automatically computed using a smooth joint model (SJM), without making any assumptions about fracture trajectories or interaction conditions. The results indicate that a higher adhesive strength of the laminated surface promotes hydraulic fracture propagation across the interface. Increasing the friction coefficient of the laminated surface from 0.15 to 0.91 resulted in a twofold increase in the fracture height. Furthermore, as the difference between vertical and horizontal principal stresses increased, the longitudinal extension distance of the fracture height significantly increased, while the activated area of the laminar surface decreased dramatically. Moreover, increasing the viscosity of the fracturing fluid led to a decrease in filtration loss along the laminar surface of the fracture and a rapid increase in net pressure, making the hydraulic fracture more likely to cross the laminar surface directly. Therefore, for heterogeneous shale oil reservoirs, a reverse-sequence fracturing technique has been proposed to enhance the length and height of the fracture. This technique involves using a high-viscosity fracturing fluid to increase the fracture height before the main construction phase, followed by a low-viscosity slickwater fracturing fluid to activate the bedding planes and promote fracture complexity. To validate the numerical modeling results, five sets of laboratory hydraulic fracturing physical simulations were conducted in Jurassic terrestrial shale. The findings revealed that as the vertical stress difference ratio increased from 0.25 to 0.6, the vertical fracture area increased by 1.98 times. Additionally, increasing both the injection displacement and the viscosity of the fracturing fluid aided in fracture height crossing of the laminar facies. These results from numerical simulation and experimental studies offer valuable insights for hydraulic fracturing design in laminated shale oil reservoirs.

Quantum computing perspective for electromagnetic wave propagation in cold magnetized plasmas

Electromagnetic waves are an inherent part of all plasmas—laboratory fusion plasmas or astrophysical plasmas. The conventional methods for studying properties of electromagnetic waves rely on discretization of Maxwell equations suitable for implementing on classical, present day, computers. The traditional methodology is not efficient for quantum computing implementation—a future computational source offering a tantalizing possibility of enormous speed up and a significant reduction in computational cost. This paper addresses two topics relevant to implementing Maxwell equations on a quantum computer. The first is on formulating a quantum Schrödinger representation of Maxwell equations for wave propagation in a cold, inhomogeneous, and magnetized plasma. This representation admits unitary, energy preserving, evolution and conveniently lends itself to appropriate discretization for a quantum computer. Riding on the coattails of these results, the second topic is on developing a sequence of unitary operators which form the basis for a qubit lattice algorithm (QLA). The QLA, suitable for quantum computers, can be implemented and tested on existing classical computers for accuracy as well as scaling of computational time with the number of available processors. In order to illustrate the QLA for Maxwell equations, results are presented from a time evolving, full wave simulation of propagation and scattering of an electromagnetic wave packet by non-dispersive dielectric medium localized in space.

Qubit lattice algorithm simulations of Maxwell’s equations for scattering from anisotropic dielectric objects

A Dyson map explicitly determines the appropriate basis of electromagnetic fields which yields a unitary representation of the Maxwell equations in an inhomogeneous medium. A qubit lattice algorithm (QLA) is then developed perturbatively to solve this representation of Maxwell equations. A QLA consists of an interleaved unitary sequence of collision operators (that entangle on lattice-site qubits) and streaming operators (that move this entanglement throughout the lattice). External potential operators are introduced to handle gradients in the refractive indices, and these operators are typically non-unitary but sparse matrices. By also interleaving the external potential operators with the unitary collide-stream operators, one achieves a QLA which conserves energy to high accuracy. Some two dimensional simulations results are presented for the scattering of a one-dimensional (1D) pulse off a localized anisotropic dielectric object.

Improved Attacks on (EC)DSA with Nonce Leakage by Lattice Sieving with Predicate

Lattice reduction algorithms have been proved to be one of the most powerful and versatile tools in public key cryptanalysis. In this work, we primarily concentrate on lattice attacks against (EC)DSA with nonce leakage via some sidechannel analysis. Previous works relying on lattice reduction algorithms such as LLL and BKZ will finally lead to the “lattice barrier”: lattice algorithms become infeasible when only fewer nonce is known. Recently, Albrecht and Heninger introduced lattice algorithms augmented with a predicate and broke the lattice barrier (Eurocrypt 2021). We improve their work in several aspects.We first propose a more efficient predicate algorithm which aims to search for the target lattice vector in a large database. Then, we combine sieving with predicate algorithm with the “dimensions for free” and “progressive sieving” techniques to further improve the performance of our attacks. Furthermore, we give a theoretic analysis on how to choose the optimal Kannan embedding factor.As a result, our algorithm outperforms the state-of-the-art lattice attacks for existing records such as 3-bit nonce leakage for a 256-bit curve and 2-bit nonce leakage for a 160-bit curve in terms of running time, sample numbers and success probability. We also break the lattice records on the 384-bit curve with 3-bit nonce leakage and the 256-bit curve with 2-bit nonce leakage which are thought infeasible previously. Finally, we give the first lattice attack against ECDSA with a single-bit nonce leakage, which enables us to break a 112-bit curve with 1-bit nonce leakage in practical time.

Effective bounds for Vinberg’s algorithm for arithmetic hyperbolic lattices

A group of isometries of a hyperbolic n-space is called a reflection group if it is generated by reflections in hyperbolic hyperplanes. Vinberg gave a semi-algorithm for finding a maximal reflection sublattice in a given arithmetic subgroup of $${{\,\textrm{O}\,}}(n,1)$$ of the simplest type. We provide an effective termination condition for Vinberg’s semi-algorithm with which it becomes an algorithm for finding maximal reflection sublattices. The main new ingredient of the proof is an upper bound for the number of faces of an arithmetic hyperbolic Coxeter polyhedron in terms of its volume.

Some comments on unitary qubit lattice algorithms for classical problems

A qubit lattice algorithm (QLA), which consists of a set of interleaved unitary collision-streaming operators, is developed for electromagnetic wave propagation in tensor dielectric media. External potential operators are required to handle gradients in the refractive indices, and these operators are typically non-unitary but sparse. A similar problem arises in the QLA for the Korteweg–de Vries equation, as the potential operator that models the KdV nonlinear term is also non-unitary. Several QLAs are presented here that avoid the need of this non-unitary potential operator by perturbing the collision operator. These QLAs are fully unitary.

The Effect of the Width of the Incident Pulse to the Dielectric Transition Layer in the Scattering of an Electromagnetic Pulse — A Qubit Lattice Algorithm Simulation

The effect of the thickness of the dielectric boundary layer that connects a material of refractive index $n_1$ to another of index $n_2$ is considered for the propagation of an electromagnetic pulse. For very thin boundary layer the scattering properties of the pulse mimics that found from the Fresnel jump conditions for a plane wave - except that the transmission to incident amplitudes are augmented by a factor of $\sqrt{n_2/n_1}$. As the boundary layer becomes thicker one finds deviations away from the Fresnel conditions and eventually one approaches WKB propagation. However there is found a small but unusual dip in part of the transmitted pulse that persists in time. The quantum lattice algorithm (QLA) used recovers the Maxwell equations to second order in a small parameter -- but QLA still recovers Maxwell equations when this parameter is unity. The expansion parameter is the speed of the pulse in medium $n_1$.

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.

Synchronous vertical propagation mechanism of multiple hydraulic fractures in shale oil formations interlayered with thin sandstone

To investigate the synchronous vertical propagation mechanism of multiple hydraulic fractures in shale oil formations interlayered with thin sandstone (SIS), we conduct a series of accurate, triaxial, hydraulic fracturing experiments using artificial SIS samples and analyze the effects of the formation dip angle and vertical stress difference on the penetration behavior of a single hydraulic fracture. Further, we develop a numerical model for the synchronous propagation of multiple fractures in SIS formations on a field scale using a three-dimensional lattice algorithm and investigate the controlling effects of the critical fracturing operation parameters on the penetration behavior of multiple hydraulic fractures. Increasing the formation dip angle inhibits the ability of the hydraulic fractures to penetrate the interlayer in the longitudinal direction significantly, while increasing the vertical, in-situ stress difference can improve this penetration ability. When the length of the segment to be fractured remains fixed, too many or few fractures are unfavorable to the longitudinal extension of the hydraulic fractures. When the segment to be fractured includes five perforation clusters, multiple hydraulic fractures on the outside can better penetrate the sandstone interlayer and enter the adjacent shale layers. Appropriately increasing the injection rate and viscosity of the fracturing fluid can enhance the penetration and extension ability of the outer hydraulic fractures; however, massive injections enhance the communication between the intermediate fractures and weaken the penetration and expansion ability of the outer fractures. An injection rate of the fracturing fluid equal to 12 m3/min can produce a better layer-penetration hydraulic-fracturing effect. When the method of intermittent pumping at a decreased injection rate of the fracturing fluid is used for fracturing under cyclic loading of first high and then low loads, the ability of multiple fractures to penetrate the interlayer in the longitudinal direction can improve further. The obtained results can provide a deeper understanding of the synchronous longitudinal propagation mechanism of multiple fractures in SIS formations, thereby providing more accurate guidance on optimizing layer-penetration fracturing parameters.

Mathematical and computer simulation of particle redistribution and inertial swarming in dispersed systems

<span>This work is devoted to modeling the motion of a dispersed phase in a gas flow in two cases: i) the dynamic characteristics of the gas flow (velocity components, stream functions and vorticity, ii) are found by numerically solving the Navier-Stokes equations with the subsequent construction of the droplet trajectory; study of the coalescence of droplets in a given model gas flow with analysis of the swarming of dispersed particles. In the first approach, the size and location of vortex and stagnant zones in a channel with a streamlined plate and with a turning flow are determined by numerically solving the equations of gas dynamics in Helmholtz variables. To study the coalescence of drops that occurs when the trajectories of two or more drops come into contact, a second approach has been developed-a computer model of inertial swarming in dispersed systems using the Delphi10 programming environment based on stochastic lattice algorithms and computable lattice enlargement. This approach is needed to pre-define the media flow rate profile. Also in this article, were analyzed the grouping and redistribution of particles for the manifestation of the "swarm of drops" effect.</span>

The irreducible vectors of a lattice:

The main idea behind lattice sieving algorithms is to reduce a sufficiently large number of lattice vectors with each other so that a set of short enough vectors is obtained. It is therefore natural to study vectors which cannot be reduced. In this work we give a concrete definition of an irreducible vector and study the properties of the set of all such vectors. We show that the set of irreducible vectors is a subset of the set of Voronoi relevant vectors and study its properties. For extremal lattices this set may contain as many as 2^n vectors, which leads us to define the notion of a complete system of irreducible vectors, whose size can be upper-bounded by the kissing number. One of our main results shows that modified heuristic sieving algorithms heuristically approximate such a set (modulo sign). We provide experiments in low dimensions which support this theory. Finally we give some applications of this set in the study of lattice problems such as SVP, SIVP and CVPP. The introduced notions, as well as various results derived along the way, may provide further insights into lattice algorithms and motivate new research into understanding these algorithms better.

Sieve algorithms for some orthogonal integer lattices

We propose in this work a Sieve algorithm that we called OrthogonalInteger sieve algorithm for some orthogonal integer lattices and particularly the case of integer lattices [Formula: see text], root lattices of type [Formula: see text] ([Formula: see text]) and of type [Formula: see text] ([Formula: see text]). In these cases, we use the famous [Formula: see text] algorithm to find the shortest vector of these lattices. Indeed, in general, a sieve algorithm builds a list of short random vectors which are not necessarily in the lattice, and try to produce short lattice vectors by taking linear combinations of the vectors in the list. But in our case, we built a list of short vectors in the lattice. From the first column of the [Formula: see text]-reduced basis of the considered basis, we have the list of at least [Formula: see text] and at most [Formula: see text] short vectors for the general case (where [Formula: see text] is the dimension of the lattice) of orthogonal integer lattices [Formula: see text] . For the lattices [Formula: see text], [Formula: see text] ([Formula: see text]) and [Formula: see text] ([Formula: see text]), we have, respectively, [Formula: see text], [Formula: see text] and [Formula: see text] short vectors. The proposed sieve algorithm for integer lattice [Formula: see text] runs in space [Formula: see text] and the OrthogonalInteger sieve algorithm performs [Formula: see text] arithmetic operations and is polynomial in space.

WITHDRAWN: Simulation and analysis of heat dissipation performance of octagonal slotted circle based on LBM lattice algorithm

This article has been withdrawn by IOP Publishing following an admission by the authors that this article may contain research errors. IOP Publishing has investigated in line with the COPE guidelines with the help of an independent subject expert, who has found the article contains numerous flaws, which bring into question the reliability of the work presented. As such, IOP Publishing agree that this work should be withdrawn.

An incremental algorithm for concept lattice based on structural similarity index

An incremental algorithm for concept lattice based on structural similarity index