Fast Computation Research Articles

Body-centered tetragonal grids for FDTD method with collocated field components and enhanced performance

Abstract We present an alternative finite-difference time-domain (FDTD) scheme that utilizes a body-centered tetragonal (BCT) grid, offering spatially collocated field components and significantly improving computational performance. The update equations for the BCT-FDTD maintain the simplicity and intuitiveness characteristic of the standard FDTD based on the Yee grid, with the key distinction being that the spatial derivative in one direction requires the use of four nearest neighbors instead of two. Our analyses of dispersion and stability reveal that the BCT-FDTD achieves a relaxed stability criterion, along with reduced anisotropy and lower phase velocity errors. Numerical tests indicate that, under equal sampling density conditions, the BCT-FDTD exhibits comparable memory usage and computation time to the standard FDTD. Moreover, our analysis establishes that when using conditions that ensure a phase velocity error of less than 1%, the BCT-FDTD requires approximately 5× less memory and achieves approximately 10× faster computation compared to standard FDTD. This improvement is expected to be even more pronounced in simulations involving anisotropic materials, thanks to the advantages provided by spatially collocated field components. Overall, we believe that the BCT-FDTD serves as a compelling alternative to the standard FDTD.

Data-Driven Improvement of Local Hybrid Functionals: Neural-Network-Based Local Mixing Functions and Power-Series Correlation Functionals.

Local hybrid functionals (LHs) use a real-space position-dependent admixture of exact exchange (EXX), governed by a local mixing function (LMF). The systematic construction of LMFs has been hampered over the years by a lack of exact physical constraints on their valence behavior. Here, we exploit a data-driven approach and train a new type of "n-LMF" as a relatively shallow neural network. The input features are of meta-GGA character, while the W4-17 atomization-energy and BH76 reaction-barrier test sets have been used for training. Simply replacing the widely used "t-LMF" of the LH20t functional by the n-LMF provides the LH24n-B95 functional. Augmented by DFT-D4 dispersion corrections, LH24n-B95-D4 remarkably improves the WTMAD-2 value for the large GMTKN55 test suite of general main-group thermochemistry, kinetics, and noncovalent interactions (NCIs) from 4.55 to 3.49 kcal/mol. As we found the limited flexibility of the B95c correlation functional to disfavor much further improvement on NCIs, we proceeded to replace it by an optimized B97c-type power-series expansion. This gives the LH24n functional. LH24n-D4 gives a WTMAD-2 value of 3.10 kcal/mol, the so far lowest value of a rung 4 functional in self-consistent calculations. The new functionals perform moderately well for organometallic transition-metal energetics while leaving room for further data-driven improvements in that area. Compared to complete neural-network functionals like DM21, the present more tailored approach to train just the LMF in a flexible but well-defined human-designed LH functional retains the possibility of graphical LMF analyses to gain deeper understanding. We find that both the present n-LMF and the recent x-LMF suppress the so-called gauge problem of local hybrids without adding a calibration function as required for other LMFs. LMF plots show that this can be traced back to large LMF values in the small-density region between the interacting atoms in NCIs for n- and x-LMFs and low values for the t-LMF. We also find that the trained n-LMF has relatively large values in covalent bonds without deteriorating binding energies. The current approach enables fast and efficient routine self-consistent calculations using n-LMFs in Turbomole.

Side-Gated Iontronic Memtransistor: A Fast and Energy-Efficient Neuromorphic Building Block.

Iontronic memtransistors have emerged as technologically superior to conventional memristors for neuromorphic applications due to their low operating voltage, additional gate control, and enhanced energy efficiency. In this study, a side-gated iontronic organic memtransistor (SG-IOMT) device is explored as a potential energy-efficient hardware building block for fast neuromorphic computing. Its operational flexibility, which encompasses the complex integration of redox activities, ion dynamics, and polaron generation, makes this device intriguing for simultaneous information storage and processing, as it effectively overcomes the von Neumann bottleneck of conventional computing. The SG-IOMT device achieves linear channel conductance performance metrics with switching speeds in the microsecond range and energy efficiency down to a few femtojoules, comparable to those of the brain. This finding demonstrates robustness, supporting the Atkinson-Shiffrin memorization model, and the four most common Hebbian learning rules. Overall, this SG-IOMT device architecture offers significant advantages over conventional architectures, as it yields remarkable image classification performance in convolutional neural network simulations.

Evaluation of dose calculation method with a combination of Monte Carlo method and removal-diffusion equation in heterogeneous geometry for boron neutron capture therapy.

Clinical research in boron neutron capture therapy (BNCT) has been conducted worldwide. Currently, the Monte Carlo (MC) method is the only dose calculation algorithm implemented in the treatment planning system for the clinical treatment of BNCT. We previously developed the MC-RD calculation method, which combines the MC method and the removal-diffusion (RD) equation, for fast dose calculation in BNCT. This study aimed to verify the partial-MC-RD calculation method, which utilizes the MC-RD calculation method for a portion of the entire neutron energy range, in terms of calculation accuracy and time as the dose calculation method. We applied the partial-MC-RD calculation method to calculate the total dose for head phantom, comprising soft tissue, brain tissue, and bone. The calculation time and accuracy were evaluated based on the full-MC method. Our accuracy verifications indicated that the partial-MC-RD calculation was mostly comparable with full-MC calculation in the accuracy. However, the assumptions and approximation used in the RD calculation mainly occurred the discrepancy from the full-MC calculation result. Additionally, the partial-MC-RD calculation reduced the time required to approximately 45% for the irradiation to the top and cheek region of head phantom, compared to the full-MC calculation. In conclusion, the MC-RD calculation method can be the basis of a fast dose calculation method in BNCT.

Pipeline ShiftAddNet: An FPGA‐Based CNN Implementation With Low Hardware Consumption Targeting Constrained Devices

ABSTRACTShiftAddNet is a recently proposed multiplier‐less CNN that replaces conventional multiplication with cheaper shift and add operations, which makes it suitable for hardware implementation. In this paper, we present the first implementation of ShiftAddNet FPGA inference core, which achieves low area consumption and fast computation. ShiftAddNet combined the convolutional layer of the DeepShift‐PS (denoted as Shift‐Accumulate, sac) and AdderNet (denoted as Add‐Accumulate, aac) into a single computational stage. Due to this reason, there are data dependencies between the sac and aac, which prohibits them from being executed in parallel, resulting in more operations compared to other multiplier‐less CNNs like DeepShift‐PS and AdderNet. To overcome this performance bottleneck, we proposed a novel technique to allow pipeline processing between sac and aac, effectively reducing the latency. The proposed ShiftAddNet‐18 was evaluated on a small ResNet‐18, achieving 11.37 ms of latency per image, which is 69.21% faster than the original version that takes 19.24 ms. On a denser network, the proposed pipeline ShiftAddNet‐101 requires only 61.92 ms as compared to the original version of 98.85 ms, showing a latency reduction of 37.1%. Compared to the state‐of‐the‐art multiplier‐less CNN core (e.g., AdderNet), our work is 20% slower in latency but provides higher accuracy and consumes less DSP.

Application of bioinformatic tools in cell type classification for single-cell RNA-seq data.

The advancements in single-cell RNA sequencing (scRNAseq) technology have significantly transformed genomics research, enabling the handling of thousands of cells in each experiment. As of now, 32,068 research studies have been cataloged in the Pubmed database. The primary aim of scRNAseq investigations is to identify cell types, understand the antitumor immune response, and identify new and uncommon cell types. Traditional techniques for identifying cell types include microscopy, histology, and pathological characteristics. However, the complexity of instruments and the need for precise experimental design make it difficult to fully capture the overall heterogeneity. Unsupervised clustering and supervised classification methods have been used to solve this task. Supervised cell type classification methods have gained popularity as large-scale, high-quality, well-annotated and more robust results compared to clustering methods. A recent study showed that support vector machine (SVM) gives a high-quality classification performance in different scenarios. In this article, we compare and evaluate the performance of four different kernels (sigmoid, linear, radial, polynomial) of SVM. The results of the experiments on three standard scRNA-seq datasets indicate that SVM with linear and SVM with sigmoid kernel classify the cells more accurately (approx. 99 %) where SVM linear kernel method has remarkably fast computation time and we also evaluate the results using some single cell specific evaluation matrices F-1 score, MCC, AUC value. Additionally, it sheds light on the potential use of kernels of SVM to give underlying information of single-cell RNA-Seq data more effectively.

THE MODEL OF CHOOSING A CRM SYSTEM IN THE MANAGEMENT OF BUSINESS ROCESSES OF SMALL BUSINESSES

The article presents a model for choosing a CRM system for small and medium-sized enterprises (SMEs), which allows for effective business processes of interaction with clients. The authors came to the conclusion that in modern conditions, digitalization of SME activities is becoming one of the pressing issues.Active digitalization of business accelerates the process of transition of the economy to digital. The digital economy is based on information and communication technologies (ICT). Information technologies in the modern economy are used for the purpose of fast, high-quality and effective computer processing of information resources, their transmission over any distance in the shortest possible time. The authors argue that government support for business digitalization cannot and should not be the only stimulating factor for SMEs. In addition to external conditions, there are also internal motivating factors - these are the capabilities of the business environment. The authors of the article substantiate the idea that CRM systems allow optimizing the process of financial and economic interaction between a company and a buyer. Using the hierarchy analysis method, the key parameters for choosing a CRM system were determined. The final indicator of the CRM system selection allows choosing the optimal CRM system for the enterprise from alternative ones. The indicator is calculated based on the second-level indicators: the functionality of the CRM system, the program interface, the tariff scale.Each of these indicators is disclosed through the indicators of the 3rd level. The weights of the indicators of all levels were determined by experts through a pairwise comparison of the elements of each level on a nine-point scale. The criteria and their weight values proposed by the authors will allow SME sector entities to choose the optimal program depending on the requests and capabilities of the enterprise.

An Efficient and Accurate Adaptive Time-Stepping Method for the Landau–Lifshitz Equation

This article presents an efficient and accurate adaptive time-stepping finite difference method (FDM) for solving the Landau–Lifshitz (LL) equation, which is an important mathematical model in understanding magnetic materials and processes. Our proposed algorithm strategically selects an adaptive time step, ensuring that the maximum displacement falls within a predefined tolerance threshold. Furthermore, this adaptive approach allows the utilization of larger time steps near equilibrium states and results in faster computations. For example, we introduce a numerical test where the adaptive time step decreases from 3.05×10−7 to 3.52×10−9. If a uniform time step is applied, around a 100 times smaller time step must be applied at unnecessary cases. To demonstrate the high performance of our proposed algorithm, we conduct several characteristic benchmark tests. The computational results confirm that the proposed algorithm is efficient and accurate. Overall, our adaptive time-stepping FDM offers a promising solution for accurately and efficiently solving the LL equation and contributes to advancements in the understanding and analysis of magnetic phenomena.

Dynamic Comfort Considerations in the Design of High-rise Buildings

<p>The following paper provides research analysis on the influence of wind on a building, considering the regional and local climate conditions, to improve dynamic stability and reduce wind disruption effects on high-rise reinforced concrete buildings. The main aspect of disturbance in living conditions is caused by wind loads on the building. The data is based on structural analysis, generated by the “Lira CAD 2013” ​​software, for a construction project of a 19-story residential building, completed in Georgia’s capital Tbilisi, 58 Kavtaradze St., in accordance with the given necessary requirements and restrictions by general terms and conditions of the Georgian state law. Computer aided modeling provides the ability to simulate and define parameters caused by the wind disruption on an actual site. Calculation process is based on variables specified by state law for each region of the country. Modified accelerations which have been caused by wind in multi-story, high-rise buildings have significantly greater values on upper floors, and reaching the edges ​​​​of the minimum living comfort criteria. The given study features the estimation for high-rise buildings under various combinations of loads. High-rise buildings have gone under significant restrictions due to use of conventional rigid frames as structural elements in the construction process. That process tends to develop a series of new research and findings for proper architectural forms, which became possible using the new generation of fast digital computing software and hardware.</p>

Predicting electronic screening for fast Koopmans spectral functional calculations

Koopmans spectral functionals are a powerful extension of Kohn-Sham density-functional theory (DFT) that enables the prediction of spectral properties with state-of-the-art accuracy. The success of these functionals relies on capturing the effects of electronic screening through scalar, orbital-dependent parameters. These parameters have to be computed for every calculation, making Koopmans spectral functionals more expensive than their DFT counterparts. In this work, we present a machine-learning model that—with minimal training—can predict these screening parameters directly from orbital densities calculated at the DFT level. We show in two prototypical use cases that using the screening parameters predicted by this model, instead of those calculated from linear response, leads to orbital energies that differ by less than 20 meV on average. Since this approach dramatically reduces run times with minimal loss of accuracy, it will enable the application of Koopmans spectral functionals to classes of problems that previously would have been prohibitively expensive, such as the prediction of temperature-dependent spectral properties. More broadly, this work demonstrates that measuring violations of piecewise linearity (i.e., curvature in total energies with respect to occupancies) can be done efficiently by combining frozen-orbital approximations and machine learning.

Identification of Physical Dynamical Processes via Linear Structure Models (Part 1)

The article discusses methods for identifying parameters of partial differential equations. Identification problems are often called poorly conditioned. However, the reason is the ambiguity of the solution, even at the point of minimality of the criterion. In particular, the article discusses: (1) An analysis of singular values for identify the unambiguous solution. The basis of these methods is a singular value decomposition of the matrix of experimental data, what makes it possible to abandon the inversion of matrices and, as a consequence, translate the problem of ill-conditioned problems into the problem of ambiguity of the solution. (2) The issues of anomalous measurements and combination of various experiments. (3) A universal optimization method for identifying parameters by their complete simple enumeration. The method is based on fast calculation of points on a multidimensional sphere. (4) The issues of identifiability of linear structure models and construction of experiments guaranteeing identification. (5) A method for identifying parameters via projecting linear structure model elements onto the plane of guarantors. (6) An approach to constructing histograms of unknown parameters of dynamic systems before calculating them using any algorithm. The approach is based on linear structure models with parameters on a sphere and a rather unexpected application of singular value decomposition. (7) The methods are accompanied by examples of heat equations. The Appendix containsalgorithmsintheMATLABlanguageforallexamples. (8)The presented optimization, projection and statistical methods based on the concept of linear structure models allow solving the same identification problem in fundamentally different ways, which significantly increases the reliability of the results obtained.

The Study on the Propagation of a Driving Laser Through Gas Target Using a Neural Network: Interaction of Intense Laser with Atoms

High-order harmonic generation is one of the ways to generate attosecond ultra-short pulses. In order to accurately simulate the high-order harmonic emission, it is necessary to perform fast and accurate calculations on the interaction between the atoms and strong laser fields. The accurate profile of the laser field is obtained from the propagation through the gas target. Under the conditions of longer wavelength driving lasers and higher gas densities, the calculation of the laser field becomes more challenging. In this paper, we utilize the driving laser electric field information obtained from numerically solving the three-dimensional Maxwell’s equations as data for machine learning, enabling the prediction of the propagation process of intense laser fields using an artificial neural network. It is found that the simulation based on frequency domain can improve the accuracy of electric field by two orders of magnitude compared with the simulation directly from time domain. On this basis, the feasibility of the transfer learning scheme for laser field prediction is further studied. This study lays a foundation for the rapid and accurate simulation of the interaction between intense laser and matter by using an artificial neural network scheme.

Review of Fast Computation Methods for Finite-State Predictive Control of Multi-Phase Drives

Optimizing the cost function in predictive control of multi-phase drives is computationally intensive. This poses a challenge since the required sampling period for drives falls within the microsecond range. Numerous methods have been proposed in the literature to address this computational demand. This paper reviews recent proposals for multi-phase drives of various kinds. A structured classification of the proposals is provided. Furthermore, an integrated framework is employed to cohesively present and connect previously unrelated methods. Key elements such as Allowed Control Set, inter-sample modulation, and weighting factor use are identified. New developments in multi-vector and single-vector approaches are discussed. Practical limitations for each approach are also considered.

Study on the Seepage Model of Three-Dimensional Vertical Well Based on the Boundary Element Method

As one of the main numerical methods, the boundary element method boasts the advantages of fast computation speed and high accuracy. Based on the considerations of the influence of pressure drop, perforation height, skin coefficient, reservoir coefficient, and reservoir radius on the seepage law, a novel three-dimensional seepage model for vertical wells was established using the Laplace transform method and Stehfest numerical inversion method. The model was utilized to examine the velocity of pressure propagation and drop amplitude, revealing that the near-well area experienced faster propagation and drop than the far-well area. During dimensional analysis, it was observed that neglecting the effects of formation pressure change could lead to a substantial increase in the dimensional error of bottom hole pressure, with a maximum error of 9.45% recorded in this study. Additionally, a sensitivity analysis was conducted on perforation length, wellbore reservoir coefficient, skin coefficient, and reservoir radius to further elucidate the seepage law of vertical wells. This study provides theoretical guidance for oil and gas production. The original contribution of this research lies in the incorporation of formation pressure changes into the three-dimensional seepage model of vertical wells for the first time, which significantly enhances the model’s accuracy and practicality.

Designing a Prototype Platform for Real-Time Event Extraction: A Scalable Natural Language Processing and Data Mining Approach

In this paper, we present a modular, high-performance prototype platform for real-time event extraction, designed to address key challenges in processing large volumes of unstructured data across applications like crisis management, social media monitoring and news aggregation. The prototype integrates advanced natural language processing (NLP) techniques (Term Frequency–Inverse Document Frequency (TF-IDF), Latent Semantic Indexing (LSI), Named Entity Recognition (NER)) with data mining strategies to improve precision in relevance scoring, clustering and entity extraction. The platform is designed to handle real-time constraints in an efficient manner, by combining TF-IDF, LSI and NER into a hybrid pipeline. Unlike the transformer-based architectures that often struggle with latency, our prototype is scalable and flexible enough to support various domains like disaster management and social media monitoring. The initial quantitative and qualitative evaluations demonstrate the platform’s efficiency, accuracy, scalability, and are validated by metrics like F1-score, response time, and user satisfaction. Its design has a balance between fast computation and precise semantic analysis, and this can make it effective for applications that necessitate rapid processing. This prototype offers a robust foundation for high-frequency data processing, adaptable and scalable for real-time scenarios. In our future work, we will further explore contextual understanding, scalability through microservices and cross-platform data fusion for expanded event coverage.

Developing A Fast Computer Vision Model for Diagnosing and Classifying Hip Fractures

Developing A Fast Computer Vision Model for Diagnosing and Classifying Hip Fractures

Modeling of a Novel T-Core Sensor with an Air Gap for Applications in Eddy Current Nondestructive Evaluation.

Multi-layer conductive structures, especially those with features like bolt holes, are vulnerable to hidden corrosion and cracking, posing a serious threat to equipment integrity. Early defect detection is vital for implementing effective maintenance strategies. However, the subtle signals produced by these defects necessitate highly sensitive non-destructive testing (NDT) techniques. Analytical modeling plays a critical role in both enhancing defect-detection capabilities and guiding the design of highly sensitive sensors for these complex structures. Compared to the finite element method (FEM), analytical approaches offer advantages, such as faster computation and high accuracy, enabling a comprehensive analysis of how sensor and material parameters influence defect detection outcomes. This paper introduces a novel T-core eddy current sensor featuring a central air gap. Utilizing the vector magnetic potential method and a truncated region eigenfunction expansion (TREE) method, an analytical model was developed to investigate the sensor's interaction with multi-layer conductive materials containing a hidden hole. The model yielded closed-form expressions for the induced eddy current density and coil impedance. A comparative study, implemented in Matlab, analyzed the eddy current distribution generated by T-core, E-core, I-core, and air core sensors under identical conditions. Furthermore, the study examined how the impedance of the T-core sensor changed at different excitation frequencies between 100 Hz and 10 kHz when positioned over a multi-layer conductor with a hidden air hole. These findings were then compared to those obtained from E-core, I-core, and air-core sensors. The analytical results were validated through finite element simulations and experimental measurements, exhibiting excellent agreement. The study further explored the influence of T-core design parameters, including the air gap radius, dome radius, core column height, and relative permeability of the T-core material, on the inspection sensitivity. Finally, the proposed T-core sensor was used to evaluate crack and hole defects in conductors, demonstrating its superior sensitivity compared to I-core and air core sensors. Although slightly less sensitive than the E-core sensor, the T-core sensor offers advantages, including a more compact design and reduced material requirements, making it well-suited for inspecting intricate and confined surfaces of the target object. This analytical model provides a valuable tool for designing advanced eddy current sensors, particularly for applications like detecting bolt hole defects or measuring the thickness of non-conductive coatings in multi-layer conductor structures.

High-Speed Sequential DNA Computing Using a Solid-State DNA Origami Register.

DNA computing leverages molecular reactions to achieve diverse information processing functions. Recently developed DNA origami registers, which could be integrated with DNA computing circuits, allow signal transmission between these circuits, enabling DNA circuits to perform complex tasks in a sequential manner, thereby enhancing the programming space and compatibility with various biomolecules of DNA computing. However, these registers support only single-write operations, and the signal transfer involves cumbersome and time-consuming register movements, limiting the speed of sequential computing. Here, we designed a solid-state DNA origami register that compresses output data from a 3D solution to a 2D surface, establishing a rewritable register suitable for solid-state storage. We developed a heterogeneous integration architecture of liquid-state circuits and solid-state registers, reducing the register-mediated signal transfer time between circuits to less than 1 h, thereby achieving fast sequential DNA computing. Furthermore, we designed a trace signal amplifier to read surface-stored signals back into solution. This compact approach not only enhances the speed of sequential DNA computing but also lays the foundation for the visual debugging and automated execution of DNA molecular algorithms.

State preparation by shallow circuits using feed forward

Fault tolerant quantum computers repetitively apply a four-step procedure: First, perform a few one and two-qubit quantum gates. Second, perform a syndrome measurement on a subset of the qubits. Third, perform fast classical computations to establish if and where errors occurred. And, fourth, correct the errors with a correction step. The next iteration applies the same procedure with new one and two-qubit gates. Even though current error-rates prohibit this procedure to work and fault tolerant quantum computing remains a distant goal, the same procedure can already prove useful today. In this work we make use of this four-step scheme not to carry out fault-tolerant computations, but to enhance short, constant-depth, quantum circuits that perform 1 qubit gates and nearest−neighbor 2 qubit gates.We introduce a new computational model called Local Alternating Quantum Classical Computations(LAQCC). In this model, qubits are placed in a grid and they can only interact with their direct neighbors; the quantum circuits are of constant depth with intermediate measurements; a classical controller can perform log-depth computations on these intermediate measurement outcomes and control future quantum operations based on the outcome. This model fits naturally between quantum algorithms in the NISQ era and full-fledged fault-tolerant quantum computation. We first prove that any Clifford circuit has an equivalent LAQCC circuit, and that any LAQCC circuit can be simulated by a QNC1circuit. Next, we conjecture the non-simulatability of LAQCC by showing that LAQCC contains the class of Instantaneous Quantum Polynomial-time circuits. We also show that any LAQCC circuit with polynomial-sized quantum circuits and unbounded classical computations is contained in the class of quantum circuits equipped with post-selection gates with respect to the task of state preparation. We continue by presenting LAQCC implementations of different subroutines, including OR-gates, quantum Fourier transforms and Threshold gates. These subroutines prove vital in constructing three state preparation routines in the main part of this work. Preparing a uniform superposition uses constant-depth arithmetic gates, combined with an exact Grover implementation by Long. For the W-state, we employ a compress-uncompress method to switch between a binary and one-hot encoding. This method extends to the more generalized Dicke-states, the superposition of n-bit strings of Hamming weight k, for k=O(n), but fails for higher k due to the birthday paradox. We extend this protocol to a protocol that prepares many-body scar states, highly excited states with low entanglement and longer coherence times than states with the same energy density. We present a circuit for preparing Dicke-states for larger k requiring log-depth circuits that maps between the factoradic number system and the combinatorial number system.

Multiplying Polynomials without Powerful Multiplication Instructions

We improve the performance of lattice-based cryptosystems Dilithium on Cortex-M3 with expensive multiplications. Our contribution is two-fold: (i) We generalize Barrett multiplication and show that the resulting shape-independent modular multiplication performs comparably to long multiplication on some platforms without special hardware when precomputation is free. We call a modular multiplication “shape-independent” if its correctness and efficiency depend only on the magnitude of moduli and not the shapes of the moduli. This was unknown in the literature even though modular multiplication has been studied for more than 40 years. In the literature, shape-independent modular multiplications often perform several times slower than long multiplications even if we ignore the cost of the precomputation. (ii) We show that polynomial multiplications based on Nussbaumer fast Fourier transform and Toom–Cook over Z2k perform the best when modular multiplications are expensive and k is not very close to the arithmetic precision.For practical evaluation, we implement assembly programs for the polynomial arithmetic used in the digital signature Dilithium on Cortex-M3. For the modular multiplications in Dilithium, our generalized Barrett multiplications are 1.92 times faster than the state-of-the-art assembly-optimized Montgomery multiplications, leading to 1.38−1.51 times faster Dilithium NTT/iNTT. Along with the improvement in accumulating products, the core polynomial arithmetic matrix-vector multiplications are 1.71−1.77 times faster. We further apply the FFT-based polynomial multiplications over Z2k to the challenge polynomial multiplication ct0, leading to 1.31 times faster computation for ct0.We additionally apply the ideas to Saber on Cortex-M3 and demonstrate their improvement to Dilithium and Saber on our 8-bit AVR environment. For Saber on Cortex-M3, we show that matrix-vector multiplications with FFT-based polynomial multiplications over Z2k are 1.42−1.46 faster than the ones with NTT-based polynomial multiplications over NTT-friendly coefficient rings. When moving to a platform with smaller arithmetic precision, such as 8-bit AVR, we improve the matrix-vector multiplication of Dilithium with our Barrett-based NTT/iNTT by a factor of 1.87−1.89. As for Saber on our 8-bit AVR environment, we show that matrixvector multiplications with NTT-based polynomial multiplications over NTT-friendly coefficient rings are faster than polynomial multiplications over Z2k due to the large k in Saber.