Linear Complexity Research Articles

ON THE BINARY SEQUENCE $(1,1,0,1,0^3,1,0^7,1,0^{15},\ldots )$

Abstract Let $\mathbb {F}$ be a field and $(s_0,\ldots ,s_{n-1})$ be a finite sequence of elements of $\mathbb {F}$ . In an earlier paper [G. H. Norton, ‘On the annihilator ideal of an inverse form’, J. Appl. Algebra Engrg. Comm. Comput.28 (2017), 31–78], we used the $\mathbb {F}[x,z]$ submodule $\mathbb {F}[x^{-1},z^{-1}]$ of Macaulay’s inverse system $\mathbb {F}[[x^{-1},z^{-1}]]$ (where z is our homogenising variable) to construct generating forms for the (homogeneous) annihilator ideal of $(s_0,\ldots ,s_{n-1})$ . We also gave an $\mathcal {O}(n^2)$ algorithm to compute a special pair of generating forms of such an annihilator ideal. Here we apply this approach to the sequence r of the title. We obtain special forms generating the annihilator ideal for $(r_0,\ldots ,r_{n-1})$ without polynomial multiplication or division, so that the algorithm becomes linear. In particular, we obtain its linear complexities. We also give additional applications of this approach.

A Supervised Ml Algorithm for Detecting and Predicting Fraud Credit Card Transactions

A Credit card fraud presents an escalating threat in today's digital economy, leading to significant financial losses for consumers, businesses, and financial institutions. Traditional detection methods are increasingly inadequate due to the rise in online transactions and the evolving complexity of fraudulent activities. This research aims to create a reliable supervised machine learning model designed to detect and predict fraudulent credit card transactions in real-time. Our approach leverages advanced algorithms and extensive datasets to enhance transaction security, thereby mitigating financial risks. Current systems, like FICO's Fraud Detection System (FDS), utilize a range of supervised learning techniques, including Decision Trees, Random Forests, and Neural Networks. However, they face limitations, including data quality issues, high false positive rates, and the need for continuous retraining to adapt to evolving fraud patterns. To address these challenges, we propose an innovative anomaly detection method based on the Isolation Forest algorithm. Unlike traditional methods that rely on distance and density measures, Isolation Forests isolate anomalies by randomly selecting features and split values, making it more efficient in identifying fraudulent transactions, especially in imbalanced datasets. The proposed system boasts low linear time complexity and minimal memory requirements, enabling the construction of high-performing models even with large datasets. The research demonstrates that the Isolation Forest approach significantly outperforms traditional models in detecting credit card fraud, offering a promising solution for enhancing security in the financial ecosystem.

Enhanced Chaotic Pseudorandom Number Generation Using Multiple Bernoulli Maps with Field Programmable Gate Array Optimizations

Certain methods for implementing chaotic maps can lead to dynamic degradation of the generated number sequences. To solve such a problem, we develop a method for generating pseudorandom number sequences based on multiple one-dimensional chaotic maps. In particular, we introduce a Bernoulli chaotic map that utilizes function transformations and constraints on its control parameter, covering complementary regions of the phase space. This approach allows the generation of chaotic number sequences with a wide coverage of phase space, thereby increasing the uncertainty in the number sequence generation process. Moreover, by incorporating a scaling factor and a sine function, we develop a robust chaotic map, called the Sine-Multiple Modified Bernoulli Chaotic Map (SM-MBCM), which ensures a high degree of randomness, validated through statistical mechanics analysis tools. Using the SM-MBCM, we propose a chaotic PRNG (CPRNG) and evaluate its quality through correlation coefficient analysis, key sensitivity tests, statistical and entropy analysis, key space evaluation, linear complexity analysis, and performance tests. Furthermore, we present an FPGA-based implementation scheme that leverages equivalent MBCM variants to optimize the electronic implementation process. Finally, we compare the proposed system with existing designs in terms of throughput and key space.

A recurrence for the surface area of (n, k)-arrangement graphs

An interesting property of an interconnected network is the number of nodes at distance i from an arbitrary processor, namely the node-centred surface area. This is an important property due to its applications in various fields of study. The ( n , k ) -arrangement graphs, denoted as A ( n , k ) are important class of graphs as they address the scalability issue in simpler topologies such as the hypercube and the star graph. Much work has been done to obtain a closed-form formula for the surface area of this class of graphs, but generally, it is not trivial to find an algorithm to compute the surface area of such graphs in polynomial time or to find an explicit formula with polynomially many terms in regard to the graph's parameters. In this paper, we present a simple recurrence that has linear computational complexity for A ( n , k ) for any arbitrary n and k in their defined range.

Single-view 3D reconstruction via dual attention

Constructing global context information and local fine-grained information simultaneously is extremely important for single-view 3D reconstruction. In this study, we propose a network that uses spatial dimension attention and channel dimension attention for single-view 3D reconstruction, named R3Davit. Specifically, R3Davit consists of an encoder and a decoder, where the encoder comes from the Davit backbone network. Different from the previous transformer backbone network, Davit focuses on spatial and channel dimensions, fully constructing global context information and local fine-grained information while maintaining linear complexity. To effectively learn features from dual attention and maintain the overall inference speed of the network, we do not use a self-attention layer in the decoder but design a decoder with a nonlinear reinforcement block, a selective state space model block, and an up-sampling Residual Block. The nonlinear enhancement block is used to enhance the nonlinear learning ability of the network. The Selective State Space Model Block replaces the role of the self-attention layer and maintains linear complexity. The up-sampling Residual Block converts voxel features into a voxel model while retaining the voxels of this layer. Features are used in the up-sampling block of the next layer. Experiments on the synthetic dataset ShapeNet and ShapeNetChairRFC with random background show that our method outperforms recent state of the art (SOTA) methods, we lead by 1% and 2% in IOU and F1 scores, respectively. Simultaneously, experiments on the real-world dataset Pix3d fully prove the robustness of our method. The code will be available at https://github.com/epicgzs1112/R3Davit.

Deep cryogenic silicon etching for 3D integrated capacitors: A numerical perspective

One promising approach to increase the capacity density of integral microcapacitors, microsupercapacitors, and microbatteries is three-dimensional structure design, where electrodes are exposed in three dimensions instead of conventional in-plane electrodes. Such structures include nanowires, nanotubes, nanopillars, nanoholes, nanosheets, and nanowalls. In this work, a cryogenic silicon etching process suitable for fabrication of structures with high electrode area is proposed. A numeric model of this process is experimentally calibrated and used for pillar array structure sidewall area optimization. The use of adaptive Runge–Kutta–Fehlberg time integrator allows to achieve almost linear overall computation complexity as a function of simulated etching time, despite the linear increase in conductance computation complexity with depth. A rule for choosing optimal geometric structure parameters under technological constraints is formulated. An optimized trefoil-like structure is proposed, resulting in a total 5.5% increase in sidewall area with respect to the hexagonal array of circular pillars, resulting in 20.33 sidewall area per unit chip area for 30 min long etch or 31.80 for 60 min long etch.

A Novel Approach for Solving the N-Queen Problem Using a Non-Sequential Conflict Resolution Algorithm

The N-Queens problem is a fundamental challenge in combinatorial optimization, commonly used as a benchmark for assessing the efficiency of algorithms. Traditional algorithms, such as Backtracking with Forward Checking (BFC), constraint satisfaction problem (CSP) techniques, Lookahead algorithms, and heuristic-based methods, often face challenges with exponential time complexity, making them less practical for large-scale instances. This paper introduces a novel algorithm, non-sequential conflict resolution (NSCR), which improves performance over traditional algorithms through dynamic conflict resolution. The NSCR algorithm iteratively resolves conflicts among queens by adjusting their positions, aiming to optimize both time complexity and memory usage. While NSCR also operates within exponential time bounds, it demonstrates improved scalability and efficiency compared to traditional methods. A significant strength of the NSCR algorithm lies in its space complexity, which is O(n), and a time complexity that, while typically lower than traditional methods, can reach O(n3) in the worst-case scenario. This linear space complexity is highly advantageous, particularly when dealing with large problem sizes, as it ensures efficient use of memory resources. Comparative analysis with the aforementioned algorithms shows that NSCR offers superior resource management, using up to 60% less memory and reducing runtime by approximately 50%, making it an efficient option for large-scale instances of the N-Queens problem. The algorithm’s performance, evaluated on problem sizes ranging from 8 to 1000 queens, highlights its ability to manage computational resources effectively, despite the inherent challenges of exponential time complexity.

An Open-Source Implementation of the Scaffold Identification and Naming System (SCINS) and Example Applications.

Organizing and partitioning sets of chemical structures is of considerable practical significance, e.g., in compound library analysis and the postprocessing of screening hit lists. Approaches such as unsupervised clustering are computationally demanding and dataset-dependent; on the other hand, rule-based methods, such as those based on Murcko scaffolds, have linear time complexity but are often too fine-grained, leading to a large number of singletons or sparsely populated classes. An alternative rule-based method that seeks to achieve an optimal balance when grouping compounds into sets is the 'Scaffold Identification and Naming System' (SCINS). To facilitate public use of this previously published method, here, we provide an open-source Python implementation of SCINS, dependent only on RDKit. We show that SCINS can be useful in identifying sparsely and densely populated regions in chemical space in large databases, here exemplified with Enamine REAL Diverse and ChEMBL. We find that Enamine REAL Diverse covers a much smaller SCINS space relative to ChEMBL, whereas the opposite is true when Murcko and generic Murcko scaffolds are considered. Additionally, we show that SCINS can result in chemically intuitive grouping of medium-sized sets of bioactive compounds, which can be useful in compound selection from virtual screening campaigns as well as postprocessing of experimental hit lists. Hence, in this work, we provide both an open-source implementation of SCINS and its characterization with relevant use cases.

Weak Quasi-Contact Metric Manifolds and New Characteristics of K-Contact and Sasakian Manifolds

Quasi-contact metric manifolds (introduced by Y. Tashiro and then studied by several authors) are a natural extension of contact metric manifolds. Weak almost-contact metric manifolds, i.e., where the linear complex structure on the contact distribution is replaced by a nonsingular skew-symmetric tensor, have been defined by the author and R. Wolak. In this paper, we study a weak analogue of quasi-contact metric manifolds. Our main results generalize some well-known theorems and provide new criterions for K-contact and Sasakian manifolds in terms of conditions on the curvature tensor and other geometric objects associated with the weak quasi-contact metric structure.

Multi-resolution visual Mamba with multi-directional selective mechanism for retinal disease detection.

Retinal diseases significantly impact patients' quality of life and increase social medical costs. Optical coherence tomography (OCT) offers high-resolution imaging for precise detection and monitoring of these conditions. While deep learning techniques have been employed to extract features from OCT images for classification, convolutional neural networks (CNNs) often fail to capture global context due to their focus on local receptive fields. Transformer-based methods, on the other hand, suffer from quadratic complexity when handling long-range dependencies. To overcome these limitations, we introduce the Multi-Resolution Visual Mamba (MRVM) model, which addresses long-range dependencies with linear computational complexity for OCT image classification. The MRVM model initially employs convolution to extract local features and subsequently utilizes the retinal Mamba to capture global dependencies. By integrating multi-scale global features, the MRVM enhances classification accuracy and overall performance. Additionally, the multi-directional selection mechanism (MSM) within the retinal Mamba improves feature extraction by concentrating on various directions, thereby better capturing complex, orientation-specific retinal patterns. Experimental results demonstrate that the MRVM model excels in differentiating retinal images with various lesions, achieving superior detection accuracy compared to traditional methods, with overall accuracies of 98.98\% and 96.21\% on two public datasets, respectively. This approach offers a novel perspective for accurately identifying retinal diseases and could contribute to the development of more robust artificial intelligence algorithms and recognition systems for medical image-assisted diagnosis.

Amortizing Circuit-PSI in the Multiple Sender/Receiver Setting

Private set intersection (PSI) is a cryptographic functionality for two parties to learn the intersection of their input sets, without leaking any other information. Circuit-PSI is a stronger PSI functionality where the parties learn only a secret-shared form of the desired intersection, thus without revealing the intersection directly. These secret shares can subsequently serve as input to a secure multiparty computation of any function on this intersection. In this paper we consider several settings in which parties take part in multiple Circuit-PSI executions with the same input set, and aim to amortize communications and computations. To that end, we build up a new framework for Circuit-PSI around generalizations of oblivious (programmable) PRFs that are extended with offline setup phases. We present several efficient instantiations of this framework with new security proofs for this setting. As a side result, we obtain a slight improvement in communication and computation complexity over the state-of-the-art semi-honest Circuit-PSI protocol by Bienstock et al. (USENIX '23). Additionally, we present a novel Circuit-PSI protocol from a PRF with secret-shared outputs, which has linear communication and computation complexity in the parties' input set sizes, and is able to realize a stronger security notion. Lastly, we derive the potential amortizations over multiple protocol executions, and observe that each of the presented instantiations is favorable in at least one of the multiple-execution settings.

MD-BiMamba: An aero-engine inter-shaft bearing fault diagnosis method based on Mamba with modal decomposition and bidirectional features fusion strategy

Inter-shaft bearing fault diagnosis can greatly save maintenance costs and guarantee the normal operation of aero-engines. However, existing methods are limited in their ability to handle sensor signals with high dimensional and complex noise. To tackle the above problems, we propose a Mamba network structure with Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) and bidirectional feature fusion for aero-engine inter-shaft bearing fault diagnosis, called MD-BiMamba. Firstly, we utilize the CEEMDAN technique to decompose the sensor signals of inter-shaft bearings and extract the intrinsic modal function by improving the noise addition method and decomposition strategy. Then, bidirectional Mamba is designed to extract the bidirectional long-range correlation of the sequence to achieve global feature extraction and fast detection of fault types with linear complexity. Moreover, an adaptive attention fusion method is proposed to achieve bidirectional feature fusion. Finally, we use the label smoothing regularization strategy to optimize the cross-entropy loss and enhance the generalization performance. Experimental results on real inter-shaft bearing datasets and noisy datasets show that the proposed method achieves 99.88 % and 95.81 % accuracy in the standard dataset and −10 dB noise environment, respectively, which is superior to other remarkable methods, in addition to achieving lower model complexity.

TS‐Mixer: A lightweight text representation model based on context awareness

AbstractLarge pre‐trained models (PTMs) have shown their powerful ability in multiple natural language processing tasks. However, using them in practical application remains a challenge due to the significant computational cost and memory requirements. In order to achieve the balance of computational cost and accuracy, MLP architecture can be used as an alternative to the self‐attention module, such as pNLP‐Mixer and Hyper‐Mixer. Experiments indicate that, MLP‐based models can attain competitive performance with low cost. They maintain the balance of computation cost and accuracy successfully, yet, this is at the expense of not being able to capture short‐range dependencies. In this paper, a novel MLP‐based model, termed TS‐Mixer, is proposed which can capture local dependencies by shifting operation. Compared with other MLP‐based models, the parameters of TS‐Mixer are decoupled from the sequence length, hence it has a smaller model size in long sequence tasks. In addition, TS‐Mixer has linear computational complexity, therefore it can be used as a lightweight alternative to the self‐attention model. Experiments show that the TS‐Mixer outperforms other MLP‐based models, which achieves higher accuracy with fewer parameters in multiple downstream tasks. Notably, compared with pre‐trained models, TS‐Mixer can reach more than 90% of their accuracy with 1% or even one thousandth of the parameters (0.174 ~ 1.2 M). These results demonstrate that TS‐Mixer can achieve a better balance between the computing resources and accuracy. Code is available at: https://github.com/wyl-privacy-project/TS-Mixer.

A lightweight visual mamba network for image recognition under resource-limited environments

The Vision Transformers (ViTs) models show great potential in recognition due to their excellent self-attention mechanisms. However, they often need more computational complexity, making achieving high performance in resource-constrained environments challenging. This paper proposes a lightweight visual model (called LightViM) to tackle these challenges, specifically devised for recognition challenges under resource-constrained environments. Considering the imperative of reducing the model’s computational complexity and resource consumption , we propose lightweight local–global feature fusion modules based on Mamba (LGF-Mamba) aimed at integrating spatially detailed local information with globally contextualized features while maintaining lower linear time complexity. In LGF-Mamba, Mamba sub-blocks are first designed to extract global information; then, for the rapid extraction of local features, LocalE module is designed and integrated into LGF-Mamba, exploring more comprehensive feature representation. This mechanism efficiently directs the network to integrate features from different spatial scales, thereby improving the recognition performance in resource-constrained environments. Experimental results indicate that, in comparison to other leading-edge lightweight methods, the proposed procedure simultaneously achieves superior recognition performance and minimal resource consumption.

Co-Optimizing Distributed Energy Resources in Linear Complexity Under Net Energy Metering

Co-Optimizing Distributed Energy Resources in Linear Complexity Under Net Energy Metering

A low-complexity error-feedback lattice-equalizer with phase tracking for underwater acoustic communications.

Recursive least squares (RLS)-based equalizers are hindered by their high complexity in underwater acoustic (UWA) communications. This article proposes an adaptive equalizer with a phase tracking method for the UWA communication, named the error-feedback lattice-equalizer (EFLE). First, we derive the algorithm for recursively solving the least squares problem from EFLE, introducing a lattice structure using time and order updates, thereby reducing the complexity to be linearly related to its length. The error-feedback mechanism used in computing reflection coefficients ensures the numerical stability of the algorithm. By focusing on the rapid tap rotation in time-varying channels, we design phase tracking in EFLE to further improve equalization performance. To verify the bit error rate (BER) performance of the proposed EFLE, we study the UWA communication system and conduct UWA simulations and at-sea experiments. Comparisons include linear complexity equalizers such as least mean square (LMS), leaky LMS, least mean mixed-norm, and ϵ-normalized LMS equalizers, and quadratic complexity RLS equalizers. At-sea experiment results show that the BER performance of EFLE significantly outperforms its counterparts.

Efficient and low-complexity variable-to-variable length coding for DNA storage

BackgroundEfficient DNA-based storage systems offer substantial capacity and longevity at reduced costs, addressing anticipated data growth. However, encoding data into DNA sequences is limited by two key constraints: 1) a maximum of h consecutive identical bases (homopolymer constraint h), and 2) a GC ratio between [0.5-cGC,0.5+cGC]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ [0.5 - c_{{GC}}, 0.5 + c_{{GC}} ] $$\\end{document} (GC content constraint cGC\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c_{GC}$$\\end{document}). Sequencing or synthesis errors tend to increase when these constraints are violated.ResultsIn this research, we address a pure source coding problem in the context of DNA storage, considering both homopolymer and GC content constraints. We introduce a novel coding technique that adheres to these constraints while maintaining linear complexity for increased block lengths and achieving near-optimal rates. We demonstrate the effectiveness of the proposed method through experiments on both randomly generated data and existing files. For example, when h=4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h = 4$$\\end{document} and cGC=0.05\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c_{GC} = 0.05$$\\end{document}, the rate reached 1.988, close to the theoretical limit of 1.990. The associated code can be accessed at GitHub.ConclusionWe propose a variable-to-variable-length encoding method that does not rely on concatenating short predefined sequences, which achieves near-optimal rates.

Enhance Key Stage Generation for Developing Kasumi Encryption Algorithm

Today, data security is a critical component of the efficient performance of every organization's diverse requirement, one of the important requirements is to provide a secure connection for data transmission in organization's networks. Cryptography algorithms are used to protect transmitted data from unauthorized access. Every day, a great deal of research is being done. The process of encrypting data is currently under development. Therefore, a substantial technological or research effort is still required. Cellular networks and secure communication. This proposal calls for the development of a new encryption system based on Using random keys in all the rounds. That’s mean we generated random keys as 128 bits for each round and save it in an array with size (8×8) because we have 8 rounds and each key divided into 8 sub keys as 16 bits for each to use in encryption and decryption. The developed random keys and traditional algorithm keys were tested with statistical tests of NIST, which made up a comparison and conclusion that the developed keys have a higher randomness specially in linear complexity, Frequency within block and random excursion tests, which leads to a better quality of encryption. Then we could test the cipher text with frequency within block test that was evaluated by (0.49590998713105255) and the approximate entropy test that was evaluated by (0.4922198616875501) for each cipher text which resulted from standard and modified Kasumi algorithm that’s means we can conclude that our modified Kasumi algorithm achieved more secure data in its resulted cipher text.

Neural Memory State Space Models for Medical Image Segmentation.

With the rapid advancement of deep learning, computer-aided diagnosis and treatment have become crucial in medicine. UNet is a widely used architecture for medical image segmentation, and various methods for improving UNet have been extensively explored. One popular approach is incorporating transformers, though their quadratic computational complexity poses challenges. Recently, State-Space Models (SSMs), exemplified by Mamba, have gained significant attention as a promising alternative due to their linear computational complexity. Another approach, neural memory Ordinary Differential Equations (nmODEs), exhibits similar principles and achieves good results. In this paper, we explore the respective strengths and weaknesses of nmODEs and SSMs and propose a novel architecture, the nmSSM decoder, which combines the advantages of both approaches. This architecture possesses powerful nonlinear representation capabilities while retaining the ability to preserve input and process global information. We construct nmSSM-UNet using the nmSSM decoder and conduct comprehensive experiments on the PH2, ISIC2018, and BU-COCO datasets to validate its effectiveness in medical image segmentation. The results demonstrate the promising application value of nmSSM-UNet. Additionally, we conducted ablation experiments to verify the effectiveness of our proposed improvements on SSMs and nmODEs.

A secured and fault-tolerant algorithm for electing an efficient coordinator in distributed networks

Selecting a reliable and stable coordinator is a crucial aspect of distributed systems, often vulnerable to node failures and security breaches. In response to these challenges, we present the Reliable and Stable Coordinator Election Algorithm in Distributed Systems (RSCEA). This algorithm is designed to withstand node failures and security threats during the election process. A key highlight of our approach is the utilization of a preference-based voting mechanism that incorporates essential nodal characteristics into the election procedure. By including nodal properties, the system's robustness is greatly increased, and a long-lasting and dependable coordinator is elected. The complexity of the RSCEA space, execution time and communication complexity were thoroughly examined. The findings show that RSCEA effectively selects a coordinator with linear message complexity proportional to the number of nodal features, denoted as O(n.m) where n is number of nodes and m indicates the number of attributes. RSCEA demonstrates scalability with a storage expense linear in number of nodes, thus markedly enhancing communication efficiency. RSCEA elects a reliable and stable coordinator in distributed networks. Its distinctive design, resilience, and effectiveness make it an important asset in the progression of distributed systems.