Multiple Complex Research Articles

On random sampling of supersingular elliptic curves

AbstractWe consider the problem of sampling random supersingular elliptic curves over finite fields of cryptographic size (SRS problem). The currently best-known method combines the reduction of a suitable complex multiplication (CM) elliptic curve and a random walk over some supersingular isogeny graph. Unfortunately, this method is not suitable when the endomorphism ring of the generated curve needs to be hidden, like in some cryptographic applications. This motivates a stricter version of the SRS problem, requiring that the sampling algorithm gives no information about the endomorphism ring of the output curve (cSRS problem). In this work we formally define the SRS and cSRS problems, which are both of theoretical interest. We discuss the relevance of the two problems for cryptographic applications, and we provide a self-contained survey of the known approaches to solve them. Those for the cSRS problem have exponential complexity in the characteristic of the base finite field (since they require computing and finding roots of polynomials of large degree), leaving the problem open. In the second part of the paper, we propose and analyse some alternative techniques—based either on the Hasse invariant or division polynomials—and we explain the reasons why they do not readily lead to efficient cSRS algorithms, but they may open promising research directions.

Laurent expansions of meromorphic modular forms

AbstractIn this paper, we study the Laurent coefficients of meromorphic modular forms at complex multiplication points by giving two approaches of computing them. The first is a generalization of the method of Rodriguez‐Villegas and Zagier, which expresses the Laurent coefficients as constant terms of a family of polynomials obtained through recursion. The second applies to meromorphic modular forms that are regularized theta lifts, and expresses their Laurent coefficients in terms of Fourier coefficients of harmonic Maass forms.

Notes on Characterizations of 2d Rational SCFTs: Algebraicity, Mirror Symmetry, and Complex Multiplication

AbstractS. Gukov and C. Vafa proposed a characterization of rational superconformal field theories (SCFTs) in dimensions with Ricci‐flat Kähler target spaces in terms of the Hodge structure of the target space, extending an earlier observation by G. Moore. The idea is refined, and a conjectural statement on necessary and sufficient conditions for such SCFTs to be rational is obtained, which is indeed proven to be true in the case the target space is . In the refined statement, the algebraicity of the geometric data of the target space turns out to be essential, and the Strominger–Yau–Zaslow fibration in the mirror correspondence also plays a vital role.

Elliptic curves with complex multiplication and abelian division fields

AbstractLet be an imaginary quadratic field, and let be the order in of conductor . Let be an elliptic curve with complex multiplication by , such that is defined by a model over , where . In this article, we classify the values of and the elliptic curves such that (i) the division field is an abelian extension of , and (ii) the ‐division field coincides with the th cyclotomic extension of the base field.

Structural and Functional Biology of Mammalian ALOX Isoforms with Particular Emphasis on Enzyme Dimerization and Their Allosteric Properties.

The human genome involves six functional arachidonic acid (AA) lipoxygenase (ALOX) genes, and the corresponding enzymes (ALOX15, ALOX15B, ALOX12, ALOX12B, ALOXE3, ALOX5) have been implicated in cell differentiations and in the pathogenesis of inflammatory, hyperproliferative, metabolic, and neurological disorders. Humans express two different AA 15-lipoxygenating ALOX isoforms, and these enzymes are called ALOX15 (15-LOX1) and ALOX15B (15-LOX2). Chromosomal localization, sequence alignments, and comparison of the enzyme properties suggest that pig and mouse ALOX15 orthologs (leukocyte-type 12-LOX) on the one hand and rabbit and human ALOX15 orthologs on the other (reticulocyte-type 15-LOX1) belong to the same enzyme family despite their different reaction specificities with AA as a substrate. In contrast, human ALOX12 (platelet-type 12-LOX), as well as pig and mouse ALOX15 (leukocyte-type 12-LOX), belong to different enzyme families, although they exhibit a similar reaction specificity with AA as a substrate. The complex multiplicity of mammalian ALOX isoforms and the controversial enzyme nomenclatures are highly confusing and prompted us to summarize the current knowledge on the biological functions, enzymatic properties, and allosteric regulation mechanisms of mammalian ALOX15, ALOX15B, and ALOX12 orthologs that belong to three different enzyme sub-families.

Prediction of Potential Suitable Distribution Areas for Northeastern China Salamander (Hynobius leechii) in Northeastern China.

The Northeastern China Salamander (Hynobius leechii) is classified as a rare, nationally protected Class II wild animal in China. Its population is declining, and its habitat is deteriorating. This study aimed to predict the distribution of suitable habitats for the Northeastern China Salamander under both current and future climate scenarios, utilizing the MaxEnt model optimized through ENMeval parameters. Species distribution data were collected from field surveys, existing literature, amphibian records in China, and the Global Biodiversity Information Network. A total of 97 records were compiled, with duplicate records within the ENMTools grid unit removed, ensuring that only one record existed within every 5 km. Ultimately, 58 distinct distribution points for the Northeastern China Salamander were identified. The R software package 'ENMeval 2.0' was employed to optimize the feature complexity (FC) and regularization multiplier (RM), and the optimized model was applied to assess the suitable distribution regions for the Northeastern China Salamander under present and future climate conditions. The findings indicated that rainfall and temperature are the primary environmental factors influencing Hynobius. Currently, the suitable habitat for the Northeastern China Salamander constitutes 6.6% of the total area of Northeastern China. Projections for the periods of 2050 and 2070 suggest that suitable habitats for the Northeastern China Salamander will continue to expand towards higher latitudes across three climate scenarios. While this study focuses solely on climate change factors and acknowledges certain limitations, it serves as a reliable reference and provides essential information for the distribution and conservation of the Northeastern China Salamander.

Short-Term and Long-Term Effects of COVID-19 and Remote Learning: Experiences of Parents Supporting Children with Mathematical Learning Disabilities in Israel

Background: Over the last three years, many studies have explored the effect of pandemic closures on learning. However, in Israel, the perspectives of parents on the short- and long-term effects of the lockdowns on students with mathematical learning disabilities (MD), have rarely been examined. Method: To fill this gap, MD (n = 33) or typically developing (n = 50) children were selected. They were in the 1st and 2nd grades during the closures, and we tested them and their surroundings, two years later. Results and Conclusions: First, according to the parent’s survey, children with MD had physical conditions similar to the TD group, the two groups had similar stable connections to the internet, computer, and a quiet environment. However, MD children (1) needed more help and (2) had a harder time concentrating during virtual math classes compared to TD children. Moreover, the coronavirus closures resulted in a greater learning gap in the MD children compared to the TD children. We found positive associations between difficulties reported by the parents and actual weakness in performances in complex multiplication and division and verbal working memory.

Unlocking the Nile Delta subsurface via advanced processing and imaging

The subsurface of the Nile Delta is infamous for challenges that are caused by a variety of geologic features that need to be resolved to unlock the full potential of the basin. The primary cause for many of the issues is the Messinian interval. This is a well-studied regional salt body with significant variation in composition, ranging from mobile shales to sand bodies over contaminated salt to anhydrite. The layer locally shows a hard top and base, which sets up mode conversions and complex multiples. The Messinian is also highly variable in terms of its geometry, thickness, faulting, and rugosity, which results in significant scattering of the seismic wavefield. The problem is further complicated by the presence of mud volcanoes that are scattered throughout the basin. Some of the mud volcanoes are currently active, and others are buried beneath more recent deposits. Free gas is often associated with these structures, and a high degree of lateral and vertical rock property variation is present, leading to complexities in seismic velocity and attenuation. The water depth varies considerably within the basin. Some producing assets exist that straddle the coastline, while others are located in water depths in excess of 1000 m. The transition is not simple due to erosional features on the seabed, leading to complex free-surface multiples. While many discoveries and developments have already been made in the post-Messinian, prospectivity in the basin is now located in the pre-Messinian, with targets ranging in depth from 2 to 10 km. When all of these features are combined, it results in a truly complex geophysical challenge.

Self-organizing hypercomplex-valued adaptive network

A novel, unsupervised, artificial intelligence system is presented, whose input signals and trainable weights consist of complex or hypercomplex values. The system uses the effect given by the complex multiplication that the multiplicand is not only scaled but also rotated. The more similar an input signal and the reference signal are, the more likely the input signal belongs to the corresponding class. The data assigned to a class during training is stored on a generic layer as well as on a layer extracting special features of the signal. As a result, the same cluster can hold a general description and the details of the signal. This property is vital for assigning a signal to an existing or a new class. To ensure that only valid new classes are opened, the system determines the variances by comparing each input signal component with the weights and adaptively adjusts its activation and threshold functions for an optimal classification decision. The presented system knows at any time all boundaries of its clusters. Experimentally, it is demonstrated that the system is able to cluster the data of multiple classes autonomously, fast, and with high accuracy.

Higher order and optimized symmetric 2D FIR filter design and hardware architecture implementation using CSD-CSE

In this paper, an optimized and high-performance Two-Dimensional (2D) Finite Impulse Response (FIR) filter is designed and hardware architecture is implemented for image processing applications. The higher-order circular symmetric 2D FIR filter is designed using a modified McClellan Transformation called a P4 transformation. The perfect circular symmetry in the contour of the 2D FIR filter and less complexity and error are attained by this proposed P4 Transformation. Next, the designed filter coefficients are represented by the Canonical Signed Digit (CSD) number format to attain the multiplier-less design to avoid the power complexity multipliers. Further, to reduce the hardware complexity, the Common Subexpression Elimination (CSE) technique is utilized to reduce the number of adders and hence area and power are decreased. Each sub-filter corresponding to the CSD coefficient rows is realized and integrated as a 2D FIR filter using a Fully Direct-type architecture. This 2D FIR filter architecture is coded by Verilog and synthesized by Cadence tools in a 45 nm CMOS technology library. The Delay, Power Consumption (PC), and Area reports were generated by the Genus synthesis tool and compared with the state-of-the-art works. The PC, area, and delay of the proposed filter architecture are reduced to the minimum of 13.96%, 69.1%, and 1.4%, respectively when compared to the existing filter architectures. The maximum of 10.48 times and a minimum of 1.18 times of Area Delay Product (ADP), and a maximum of 10.69 times and a minimum of 1.063 times of Power Delay Product (PDP) are smaller than the existing 2D filter architectures.

Evaluating the Efficiency of zk-SNARK, zk-STARK, and Bulletproof in Real-World Scenarios: A Benchmark Study

This study builds on our previous systematic literature review (SLR) that assessed the applications and performance of zk-SNARK, zk-STARK, and Bulletproof non-interactive zero-knowledge proof (NIZKP) protocols. To address the identified research gaps, we designed and implemented a benchmark comparing these three protocols using a dynamic minimized multiplicative complexity (MiMC) hash application. We evaluated performance across four general-purpose programming libraries and two programming languages. Our results show that zk-SNARK produced the smallest proofs, while zk-STARK generated the largest. In terms of proof generation and verification times, zk-STARK was the fastest, and Bulletproof was the slowest. Interestingly, zk-SNARK proofs verified marginally faster than zk-STARK, contrary to other findings. These insights enhance our understanding of the functionality, security, and performance of NIZKP protocols, providing valuable guidance for selecting the most suitable protocol for specific applications.

Toward Finding S-Box Circuits With Optimal Multiplicative Complexity

Toward Finding S-Box Circuits With Optimal Multiplicative Complexity

Construction of lattices over the real sub-field of ℚ(ς8) for block fading (wiretap) coding

In this paper, we consider the challenge of ensuring secure communication in block-fading wiretap channels using encoding techniques. We investigate the coding for block-fading wiretap channels using stacked lattice codes constructed over completely real number fields, which is a well-established technique. In this paper, we consider degree-four complex multiplication field [Formula: see text] over [Formula: see text]. We employ binary codes to generate a lattice over [Formula: see text], which is subsequently used to form an integral lattice. The resulting integral lattice can be effectively applied to enhance the security of communication within block-fading wiretap channels.

Low-complexity two-stage carrier phase recovery using principal component reconstruction analysis and optimal decision maximum likelihood for PS-MQAM systems.

A low-complexity two-stage carrier phase recovery (CPR) scheme based on principal component reconstruction analysis and optimal decision maximum likelihood (PCRA-OML) is proposed for probabilistic shaping (PS)-M quadrature amplitude modulation (QAM) coherent optical communication systems. In the first stage, symbols on the QPSK-shaped ring are selected, their amplitude noise is eliminated, and their amplitudes are increased, thus completing the reconstruction of the principal component. Lastly, the carrier phase is recovered using principal component phase estimation (PCPE). In the second stage, a maximum likelihood phase estimation algorithm based on optimal decision-making is employed to further enhance the stability and robustness of the scheme. The effectiveness of the proposed scheme is validated through 28 GBaud polarization division multiplexing PS-64QAM simulations and 28 GBaud PS-16QAM experiments. The experimental results indicate that in the PS-16QAM entropy of 3 bit/symbol system, when the threshold for normalized generalized mutual information is 0.9, the proposed scheme provides a 1.7 dB optical signal-to-noise ratio (OSNR) gain compared to blind phase search (BPS), and an additional 0.9 dB OSNR gain compared to the PCPE scheme. The proposed PCRA-OML scheme exhibits versatility across various shaping strengths and is less susceptible to probabilistic influences than the BPS and PCPE schemes. Additionally, under the premise of comparable performance in the PS-64QAM system, the proposed scheme reduces over 90% of the computational complexity associated with multiplication compared to BPS. In the PS-16QAM system, the proposed scheme's real multiplication complexity is only 17.8% of BPS, achieving an overall O(N) complexity.

Nonvanishing of L-function of some Hecke characters on cyclotomic fields

In this paper, we show the nonvanishing of some Hecke characters on cyclotomic fields. The main ingredient of this paper is a computation of eigenfunctions and the action of Weil representation at some primes including the primes above 2. As an application, we show that for each isogeny factor of the Jacobian of the p-th Fermat curve where 2 is a quadratic residue modulo p, there are infinitely many twists whose analytic rank is zero. Also, for a certain hyperelliptic curve over the 11-th cyclotomic field whose Jacobian has complex multiplication, there are infinitely many twists whose analytic rank is zero.

Deep Learning-Driven Compiler Enhancements for Efficient Matrix Multiplication

Matrix multiplication is a fundamental operation in many computational fields, requiring optimization to handle increasing data sizes efficiently. In this paper, the implementation of Deep Learning in Matrix multiplication is reviewed, which is considered important nowadays due to the growing complexity of matrix multiplication for gaming and complex programs. The current standard matrix multiplication and the time taken by it on different matrix sizes are described. The Tiled Matrix multiplication, which trims the matrix into various pieces and calculates the product for each piece, and thereafter combines the result, is also described. The times taken by both methods for different matrix sizes were compared. The main idea was to use Deep Neural Networks (DNN) to compare and rank code variants that are obtained in pieces and determine their relative performance. A tournament-based ranking system is used for assigning ranks to the code versions. The effectiveness of these techniques was evaluated on various matrix multiplication operations commonly found in deep learning workloads. Up to 8.844x speedup over the naive implementation for a matrix size of 1024 is achieved by this approach. The results demonstrate the effectiveness of combining compiler optimization techniques and deep learning models in optimizing matrix multiplication.

The Mechanisms of Adipose Stem Cell-Derived Exosomes Promote Wound Healing and Regeneration.

Chronic wound healing is a class of diseases influenced by multiple complex factors, causing severe psychological and physiological impact on patients. It is an intractable clinical challenge and its possible mechanisms are not yet clear. It has been proven that adipose stem cell-derived exosomes (ADSC-Exos) can promote wound healing and inhibit scar formation by regulating inflammation, promoting cell proliferation, migration, and angiogenesis, regulating matrix remodeling, which provides a new approach for wound healing through biological treatment. This review focuses on the mechanism, treatment, and administration methods of ADSC-Exos in wound healing, providing a comprehensive understanding the mechanisms of ADSC-Exos on wound healing. LEVEL OF EVIDENCE I: This journal requires that authors assign a level of evidence to each article. For a full description of these Evidence-Based Medicine ratings, please refer to the Table of Contents or the online Instructions to Authors www.springer.com/00266 .

The tensor as an informational resource.

A tensor is a multidimensional array of numbers that can be used to store data, encode a computational relation, and represent quantum entanglement. In this sense, a tensor can be viewed as valuable resource whose transformation can lead to an understanding of structure in data, computational complexity, and quantum information. In order to facilitate the understanding of this resource, we propose a family of information-theoretically constructed preorders on tensors, which can be used to compare tensors with each other and to assess the existence of transformations between them. The construction places copies of a given tensor at the edges of a hypergraph and allows transformations at the vertices. A preorder is then induced by the transformations possible in a given growing sequence of hypergraphs. The new family of preorders generalizes the asymptotic restriction preorder which Strassen defined in order to study the computational complexity of matrix multiplication. We derive general properties of the preorders and their associated asymptotic notions of tensor rank and view recent results on tensor rank nonadditivity, tensor networks, and algebraic complexity in this unifying frame. We hope that this work will provide a useful vantage point for exploring tensors in applied mathematics, physics, and computer science but also from a purely mathematical point of view.

P∞$p^{\infty }$‐Selmer ranks of CM abelian varieties

AbstractFor an elliptic curve with complex multiplication over a number field, the ‐Selmer rank is even for all . Česnavičius proved this using the fact that admits a ‐isogeny whenever splits in the complex multiplication field, and invoking known cases of the ‐parity conjecture. We give a direct proof, and generalise the result to abelian varieties.

The Birch-Swinnerton-Dyer conjecture of elliptic curves over mathbbQwith complex multiplication

The Birch-Swinnerton-Dyer conjecture of elliptic curves over mathbbQwith complex multiplication