Search Results (21)

The rapid advancement of quantum computing poses a severe threat to traditional public key cryptosystems. Lattice-based cryptography has emerged as a core candidate for post-quantum cryptography due to its presumed quantum resistance, robust security foundations, and functional versatility, with its concrete security relying on the computational hardness of lattice problems. Existing lattice-based cryptography surveys mainly focus on cryptosystem design, scheme comparisons, and post-quantum cryptography standardization progress, with only cursory coverage of classical lattice algorithms that underpin the concrete security of lattice-based cryptography. We present the first systematic survey of classical lattice algorithms, focusing on two core categories of algorithms for solving lattice problems: approximate algorithms and exact algorithms. The approximate algorithms cover mainstream lattice basis reduction methods such as Lenstra–Lenstra–Lovász (LLL), Block Korkine–Zolotarev (BKZ), and General Sieve Kernel (G6K) algorithms, as well as alternative frameworks. The exact algorithms encompass dominant techniques like enumeration and sieving algorithms, along with alternative strategies. We systematically trace the evolutionary trajectory and inherent logical connections of various algorithms, clarify their core mechanisms, and identify promising future research directions. This survey not only serves as an introductory guide for beginners but also provides a valuable reference for seasoned researchers, facilitating the concrete security evaluation of lattice-based cryptosystems and the design of novel lattice algorithms. Full article

We study double-circulant codes over a class of semi-local rings arising from the idempotent construction $R={\mathbb{Z}}_{p^2}+u{\mathbb{Z}}_{p^2}$, where $u^2=u$, and p is an odd prime. Although both algebraic settings considered admit this presentation, they correspond to two distinct rings depending on whether the additional relation $pu=0$ is imposed or not. These two configurations induce different ideal lattices and symmetry properties, which play a decisive role in the structure and enumeration of codes. Exploiting the Chinese Remainder Theorem, we describe self-dual and linear complementary dual (LCD) double-circulant codes in a unified, componentwise manner. Exact enumeration formulas are derived by reducing the corresponding duality constraints to norm equations over finite fields and unramified Galois extensions of ${\mathbb{Z}}_{p^2}$. We further construct explicit ${\mathbb{F}}_p$-linear Gray maps from $R^{2n}$ to ${\mathbb{F}}_p^{6n}$ in the degenerate case $pu=0$ and to ${\mathbb{F}}_p^{8n}$ in the standard case $pu\ne 0$, and show that these maps preserve self-duality and the LCD property. Assuming a standard primitive-root hypothesis on the code length, as predicted by Artin’s primitive root conjecture, we establish asymptotic existence bounds for the Gray images of both LCD and self-dual double-circulant codes via a probabilistic argument. The degenerate case $pu=0$ yields a shorter Gray expansion and a stronger self-dual entropy threshold, while the case $pu\ne 0$ leads to a larger self-dual ensemble with distinct asymptotic characteristics. Full article

We present a fully controlled thermodynamic study of the two-dimensional dipolar Q-state clock model on small square lattices with free boundaries, combining exhaustive state enumeration with noise-free evaluation of canonical observables. We resolve the complete energy spectra and degeneracies $\{E_n,c_n\}$ for the Ising case ($Q=2$) on lattices of size $L=3,4,5$, and for clock symmetries $Q=4,6,8$ on a $3\times 3$ lattice, tracking how the competition between exchange and long-range dipolar interactions reorganizes the low-energy manifold as the ratio $\alpha =D/J$ is varied. Beyond a finite-size characterization, we identify several qualitatively new thermodynamic signatures induced solely by dipolar anisotropy. First, we demonstrate that ground-state level crossings generated by long-range interactions appear as exact zeros of the specific heat in the limit $C(T\to 0,\alpha )$, establishing an unambiguous correspondence between microscopic spectral rearrangements and macroscopic caloric response. Second, we show that the shape of the associated Schottky-like anomalies encodes detailed information about the degeneracy structure of the competing low-energy states: odd lattices ($L=3,5$) display strongly asymmetric peaks due to unbalanced multiplicities, whereas the even lattice ($L=4$) exhibits three critical values of $\alpha$ accompanied by nearly symmetric anomalies, reflecting paired degeneracies and revealing lattice parity as a key organizing principle. Third, we uncover a symmetry-driven crossover with increasing Q: while the $Q=2$ and $Q=4$ models retain sharp dipolar-induced critical points and pronounced low-temperature structure, for $Q\ge 6$, the energy landscape becomes sufficiently smooth to suppress ground-state crossings altogether, yielding purely thermal specific-heat maxima. Altogether, our results provide a unified, size- and symmetry-resolved picture of how long-range anisotropy, lattice parity, and discrete rotational symmetry shape the thermodynamics of mesoscopic magnetic systems. We show that dipolar interactions alone are sufficient to generate nontrivial critical-like caloric behavior in clusters as small as $3\times 3$, establishing exact finite-size benchmarks directly relevant for van der Waals nanomagnets, artificial spin-ice arrays, and dipolar-coupled nanomagnetic structures. Full article

For a graph G of a degree greater than or equal to 3, counting the number of independent sets (denoted as $i\left(G\right)$) is a classical #P-complete problem. Here, we establish a new worst-case upper bound time complexity for computing $i\left(G\right)$ for any non-constraint undirected graph. Our proposal applies the vertex division rule $i\left(G\right)=i(G-\{x\left\}\right)+i(G-N[x\left]\right)$ over a vertex x which satisfies some conditions, and considers cactus and outerplanar graphs as basic subgraphs.Our algorithm establishes a leading worst-case upper bound of $O^{*}(1.{2321}^n)$, where n is the number of vertices in the graph and $O^{*}$ omits polynomial terms in n. Full article

The spin-one Ising model on the honeycomb lattice has never been solved exactly in spite of its simplicity. Even its exact critical temperature is not known. The exact integer values for the density of states of the spin-one Ising model on the $L\times 2L$ honeycomb lattice are enumerated up to $L=14$. The partition function zeros in the complex temperature plane of the spin-one Ising model on the $L\times 2L$ honeycomb lattice are exactly obtained, using the density of states. The properties of the partition function zeros in the complex temperature plane are related to the behaviors of various thermodynamic functions, in particular, their singular behaviors. The unknown properties of the spin-one Ising model on the honeycomb lattice are investigated, based on its partition function zeros in the complex temperature plane. Full article

In graph theory and network design, the minimum cut is a fundamental measure of system connectivity and communication capacity. While prior research has largely focused on computing the minimum cut for a fixed source–sink pair, practical scenarios such as data center communication often demand a different objective: identifying the source node whose minimum cut to a designated sink is maximized. This task, which we term the Global Maximum Minimum Cut with Fixed Sink (GMMC-FS) problem, captures the goal of locating a high-capacity source relative to a shared sink node that aggregates multiple servers. The problem is of significant engineering importance, yet it is computationally challenging as it involves a nested max–min optimization. In this paper, we present a recursive reduction (RR) algorithm for solving the GMMC-FS problem. The key idea is to iteratively select pivot nodes, compute their minimum cuts with respect to the sink, and prune dominated candidates whose cut values cannot exceed that of the pivot. By recursively applying this elimination process, RR dramatically reduces the number of max-flow computations required while preserving exact correctness. Compared with classical contraction-based and Gomory–Hu tree approaches that rely on global cut enumeration, the proposed RR framework offers a more direct and scalable mechanism for identifying the source that maximizes the minimum cut to a fixed sink. Its novelty lies in exploiting the structural properties of the sink side of suboptimal cuts, which leads to both theoretical efficiency and empirical robustness across large-scale networks. We provide a rigorous theoretical analysis establishing both correctness and complexity bounds, and we validate the approach through extensive experiments. Results demonstrate that RR consistently achieves optimal solutions while significantly outperforming baseline methods in runtime, particularly on large and dense networks. Full article

In this paper, we investigate the cycle structure inherent in the Tanner graphs of low-density parity-check (LDPC) codes constructed from balanced incomplete block designs (BIBDs). We begin by delineating the incidence structure of BIBDs and propose a methodology for constructing LDPC codes based on these designs. By analyzing the incidence relations between points and blocks within a BIBD, we prove that the resulting LDPC codes possess a girth of 6. Subsequently, we provide a detailed analysis of the cycle structure of the constructed LDPC codes and introduce a systematic approach for enumerating their short cycles. Using this method, we determine the exact numbers of cycles of lengths 6 and 8. Simulation results demonstrate that the constructed LDPC codes exhibit excellent performance. Full article

Background: Understanding the extent of the disease penetration and assessing its impact is critical during a pandemic. However, laboratory-based COVID-19 estimation can be resource-intensive and may not be feasible during an emergency, particularly in low-resource settings. Aim: To investigate whether self-reported symptoms can be used for COVID-19 screening to estimate the burden among individuals aged 18 years and above in a rural setting. Methods: A community-based cross-section study was conducted in a rural district of Haryana, a state in north India, using a self-reported semi-structured questionnaire developed on a digital platform. Information on COVID-19 manifestations as essential and non-essential, confirmed laboratory tests, and disability data using Washington Groups of Short Set were obtained. The sensitivity of the COVID-19 symptoms was estimated against laboratory-confirmed true positives. A chi-square or Fisher exact test for association and a multivariable regression to determine the predictors of the prevalence was carried out. Results: In total, 2954 respondents (79.8%), out of 3700 enumerated, were interviewed. The mean age of respondents was 42 years (SD 17.2), with 54.8% female respondents. The prevalence of COVID-19 based on self-reported symptoms was 6.2% (95%CI: 5.3–7.1). The age-adjusted prevalence was 6.04% (95%CI: 5.9–6.1). Of the total COVID-19 cases, 170 (5.7%, 95%CI: 4.9–6.5) revealed a laboratory-confirmed test. Given three essential symptoms to declare provisionally COVID-19 cases, the sensitivity was 82.9% (141/170), but considering two or more essential symptoms along with two or more non-essential, the sensitivity reached up to 91.8% (156/170). The multivariable analysis showed that increased age, higher education attainment, students, entrepreneurs, persons working in private sectors, and participants with poor hygiene were predictors. Conclusions: A symptoms-based identification of COVID-19 cases can give a reliable estimate and valuable insight into the extent of the penetration, especially in low-middle-income countries, and can be a supplement, not a replacement, to a laboratory test. Full article

Discrete statistical systems offer a significant advantage over systems defined in the continuum, since they allow for an easier enumeration of microstates. We introduce a lattice-gas model on the vertices of a polyhedron called a pentakis icosidodecahedron and draw its exact phase diagram by the Wang–Landau method. Using different values for the couplings between first-, second-, and third-neighbor particles, we explore various interaction patterns for the model, ranging from softly repulsive to Lennard-Jones-like and SALR. We highlight the existence of sharp transitions between distinct low-temperature “phases”, featuring, among others, regular polyhedral, cluster-crystal-like, and worm-like structures. When attempting to reproduce the equation of state of the model by Monte Carlo simulation, we find hysteretic behavior near zero temperature, implying a bottleneck issue for Metropolis dynamics near phase-crossover points. Full article

A safe water supply is a necessity, but it remains one of the backlogs of services rendered in rural areas of developing countries. This leads to vulnerable communities using water from available sources that is unsafe as it is contaminated with faecal matter. Microbial source tracking (MST) methods are gold-standard techniques that detect the exact sources of faecal contamination. This study, therefore, tracked and identified the exact sources of faecal contamination from the catchment to the point of use in rural areas of Vhembe District Municipality. Collected water samples (n = 1048) were concentrated by membrane filtration for the enumeration and detection of E. coli, followed by DNA extraction. The extracted DNA was subjected to a quantitative polymerase chain reaction (qPCR) to track target host-specific Bacteroidales genetic markers from the water source to the point of use. Rivers and dams exhibited maximum E. coli counts of up to 90 CFU/100 mL during the wet season and up to 50 CFU/100 mL during the dry season. Due to the effective treatment of these water sources, no E. coli bacteria were detected in any of the sampled municipal drinking water treatment plants at the point of treatment, while this indicator bacterium was detected at the point of use (households), with a maximum of 4 CFU/100 mL recorded during both the wet and dry seasons. Overall, the most prevalent MST marker exhibited during the wet season was BacCan (dog-associated, 6.87%), followed by BacCow (cow-associated, 5.53%), while Pig-2-Bac (pig-associated, 2.48%) was the least prevalent. The most prevalent marker exhibited during the dry season was BacCan (5.34%), followed by BacCow, with Pig-2-Bac (1.72%) being the least prevalent. A positive correlation (r = 0.31, p = 0.001) was established between the presence of the MST markers and detected E. coli from water sources to the point of use. The knowledge of the faecal contamination attributes in both public and domestic domains will assist in developing prevention and control strategies. Full article

Descriptive geometry has indispensable applications in many engineering activities. A summary of these is provided in the first chapter of this paper, preceded by a brief introduction into the methods of representation and mathematical recognition related to our research area, such as projection perpendicular to a single plane, projection images created by perpendicular projection onto two mutually perpendicular image planes, but placed on one plane, including the research of curves and movements, visual representation and perception relying on a mathematical approach, and studies on toothed driving pairs and tool geometry in order to place the development presented here among them. As a result of the continuous variability of the technological environment according to various optimization aspects, the engineering activities must also be continuously adapted to the changes, for which an appropriate approach and formulation are required from the practitioners of descriptive geometry, and can even lead to improvement in the field of descriptive geometry. The imaging procedures are always based on the methods and theorems of descriptive geometry. Our aim was to examine the spatial variation in the wear of the tool edge and the machining of the components of toothed drive pairs using two cameras. Resolving contradictions in spatial geometry reconstruction research is a constant challenge, to which a possible answer in many cases is the searching for the right projection direction, and positioning cameras appropriately. A special method of enumerating the possible infinite viewpoints for the reconstruction of tool surface edge curves is presented in the second part of this paper. In the case of the monitoring the shape geometry, taking into account the interchangeability of the projection directions, i.e., the property of symmetry, all images made from two perpendicular directions were taken into account. The procedure for determining the correct directions in a mathematically exact way is also presented through examples. A new criterion was formulated for the tested tooth edge of the hob to take into account the shading of the tooth next to it. The analysis and some of the results of the Monge mapping, suitable for the solution of a mechanical engineering task to be solved in a specific technical environment, namely defining the conditions for camera placements that ensure reconstructibility are also presented. Taking physical shadowing into account, conclusions can be drawn about the degree of distortion of the machined surface from the spatial deformation of the edge curve of the tool reconstructed with correctly positioned cameras. Full article

Machine scheduling is a hard combinatorial problem having many manifestations in real life. Due to the schedule followed, the possibility of installations of machines operating sub-optimally is high. In this work, we examine the problem of a single machine with time-dependent capacity that performs jobs of deterministic durations, while for each job, its due time is known in advance. The objective is to minimize the aggregated tardiness in all tasks. The problem was motivated by the need to schedule charging times of electric vehicles effectively. We formulate an integer programming model that clearly describes the problem and a constraint programming model capable of effectively solving it. Due to the usage of interval variables, global constraints, a powerful constraint programming solver, and a heuristic we have identified, which we call the “due times rule”, the constraint programming model can reach excellent solutions. Furthermore, we employ a hybrid approach that exploits three local search improvement procedures in a schema where the constraint programming part of the solver plays a central role. These improvement procedures exhaustively enumerate portions of the search space by exchanging consecutive jobs with a single job of the same duration, moving cost-incurring jobs to earlier times in a consecutive sequence of jobs or even exploiting periods where capacity is not fully utilized to rearrange jobs. On the other hand, subproblems are given to the exact constraint programming solver, allowing freedom of movement only to certain parts of the schedule, either in vertical ribbons of the time axis or in groups of consecutive sequences of jobs. Experiments on publicly available data show that our approach is highly competitive and achieves the new best results in many problem instances. Full article

We compute the exact root-mean-square end-to-end distance of the interacting self-avoiding walk (ISAW) up to 27 steps on the simple cubic lattice. These data are used to construct a fixed point equation to estimate the theta temperature of the collapse transition of the ISAW. With the Bulirsch–Stoer extrapolation method, we obtain accurate results that can be compared with large-scale long-chain simulations. The free parameter $\omega$ in extrapolation is precisely determined using a parity property of the ISAW. The systematic improvement of this approach is feasible by adopting the combination of exact enumeration and multicanonical simulations. Full article

Fusarium head blight (FHB) is one of the most economically destructive diseases of wheat (Triticum aestivum L.), causing substantial yield and quality loss worldwide. Fusarium graminearum is the predominant causal pathogen of FHB in the U.S., and produces deoxynivalenol (DON), a mycotoxin that accumulates in the grain throughout infection. FHB results in kernel damage, a visual symptom that is quantified by a human observer enumerating or estimating the percentage of Fusarium-damaged kernels (FDK) in a sample of grain. To date, FDK estimation is the most efficient and accurate method of predicting DON content without measuring presence in a laboratory. For this experiment, 1266 entries collectively representing elite varieties and SunGrains advanced breeding lines encompassing four inoculated FHB nurseries were represented in the analysis. All plots were subjected to a manual FDK count, both exact and estimated, near-infrared spectroscopy (NIR) analysis, DON laboratory analysis, and digital imaging seed phenotyping using the Vibe QM3 instrument developed by Vibe imaging analytics. Among the FDK analytical platforms used to establish percentage FDK within grain samples, Vibe QM3 showed the strongest prediction capabilities of DON content in experimental samples, R² = 0.63, and higher yet when deployed as FDK GEBVs, R² = 0.76. Additionally, Vibe QM3 was shown to detect a significant SNP association at locus S3B_9439629 within major FHB resistance quantitative trait locus (QTL) Fhb1. Visual estimates of FDK showed higher prediction capabilities of DON content in grain subsamples than previously expected when deployed as genomic estimated breeding values (GEBVs) (R² = 0.71), and the highest accuracy in genomic prediction, followed by Vibe QM3 digital imaging, with average Pearson’s correlations of r = 0.594 and r = 0.588 between observed and predicted values, respectively. These results demonstrate that seed phenotyping using traditional or automated platforms to determine FDK boast various throughput and efficacy that must be weighed appropriately when determining application in breeding programs to screen for and develop resistance to FHB and DON accumulation in wheat germplasms. Full article

NP-complete problems in graphs, such as enumeration and the selection of subgraphs with given characteristics, become especially relevant for large graphs and networks. Herein, particular statements with constraints are proposed to solve such problems, and subclasses of graphs are distinguished. We propose a class of prefractal graphs and review particular statements of NP-complete problems. As an example, algorithms for searching for spanning trees and packing bipartite graphs are proposed. The developed algorithms are polynomial and based on well-known algorithms and are used in the form of procedures. We propose to use the class of prefractal graphs as a tool for studying NP-complete problems and identifying conditions for their solvability. Using prefractal graphs for the modeling of large graphs and networks, it is possible to obtain approximate solutions, and some exact solutions, for problems on natural objects—social networks, transport networks, etc. Full article