Computation Of Polynomials Research Articles

Prediction of cluster energy of lithium clusters from codescriptors with application in 2D porous graphene

A lithium metal anode has a high energy density, but its applications face numerous challenges, particularly the irregular deposition of lithium dendrites, which form due to the aggregation of lithium clusters, posing serious safety risks. Research has revealed that small lithium clusters have the potential to enhance the electrode potential in Lithium-ion batteries (LIBs), and cluster energy affects the stability of lithium clusters. Modern computational tools and DFT calculations of lithium clusters can provide invaluable insights into their chemical, physical, and structural properties. However, calculating structural properties through large-scale experiments can be time-consuming and expensive. To overcome the challenges inherent in studying lithium clusters, this study employs a novel approach. We construct molecular graphs for the most stable structures and leverage CoM Polynomial computations to generate codescriptors, enabling efficient and accurate predictions of cluster energy for diverse lithium cluster configurations. The curvilinear regression analysis filters out highly significant regression equations and highly predictive codescriptors. Additionally, we investigated two molecular structures of porous graphene, namely linear and triangular, and obtained topological codescriptors’ analytical expressions by utilizing the CoM Polynomial in each case. A deeper understanding of both lithium clusters and porous graphene properties is essential for optimizing LIBs’ performance and stability.

Computation of polynomial and rational approximations in complex domains by the $$\tau $$-method

Computation of polynomial and rational approximations in complex domains by the $$\tau $$-method

Privacy-preserving fair outsourcing polynomial computation without FHE and FPR

With the development of cloud computing, verifiable outsourcing computation (VC) has received much more attention. The polynomial is a fundamental mathematical function with widespread applications. Plenty of VC schemes for polynomials have been proposed recently. However, most previous schemes focus on ensuring that the client can get a valid result returned by the cloud service provide (CSP) before payment, while often ignoring the CSP’s interest. To the best of our knowledge, Guan et al. (2021) proposed a pioneering framework for building fair outsourcing polynomial computation, which serves as the state of the art. However, it discloses the privacy of outsourced polynomials, inputs, and outputs. Furthermore, it suffers from a false positive rate (FPR) in the verification phase due to the sampling technique. As a result, it breaks the fairness between the client and the CSP.To solve these problems, we propose a privacy-preserving fair outsourcing polynomial computation without FPR. To avoid expensive Fully Homomorphic Encryption (FHE), we utilize Paillier encryption and blind technique to ensure privacy. Our proposed scheme can guarantee fairness with an overwhelming probability by applying the SGX technique. The comprehensive performance evaluation and extensive simulations show that our protocol is more practical in cloud computing.

ESTARK: extending STARKs with arguments

STARK is a widely used transparent proof system that uses low-degree tests for proving the correctness of a computer program. STARK consumes an intermediate representation known as AIR that is more appropriate for programs with a relatively short and structured description. However, an AIR is not able to succinctly express non-equality constraints, leading to the incorporation of unwanted polynomials. We present the eSTARK protocol, a new probabilistic proof that generalizes the STARK family through the introduction of a more generic intermediate representation called eAIR. We describe eSTARK in the polynomial IOP model, which combines the optimized version of the STARK protocol with the incorporation of three arguments into the protocol. We also explain various techniques that enhance the vanilla STARK complexity, including optimizations applied to polynomial computations, and analyze the tradeoffs between controlling the constraint degree either at the representation of the AIR or inside the eSTARK itself.

SoK: Fully Homomorphic Encryption Accelerators

Fully Homomorphic Encryption (FHE) is a key technology enabling privacy-preserving computing. However, the fundamental challenge of FHE is its inefficiency, due primarily to the underlying polynomial computations with high computation complexity and extremely time-consuming ciphertext maintenance operations. To tackle this challenge, various FHE accelerators have recently been proposed by both research and industrial communities. This paper takes the first initiative to conduct a systematic study on the 14 FHE accelerators — cuHE/cuFHE, nuFHE, HEAT, HEAX, HEXL, HEXL-FPGA, 100 ×, F1, CraterLake, BTS, ARK, Poseidon, FAB and TensorFHE. We first make our observations on the evolution trajectory of these existing FHE accelerators to establish a qualitative connection between them. Then, we perform testbed evaluations of representative open-source FHE accelerators to provide a quantitative comparison on them. Finally, with the insights learned from both qualitative and quantitative studies, we discuss potential directions to inform the future design and implementation for FHE accelerators.

Algorithms for computing chromatic polynomials and chromatic index polynomials

ObjectivesThe aim of this article is to enhance the understanding of the computation of chromatic polynomials and chromatic index polynomials, and to facilitate their practical use in various fields by demonstrating and supporting the proposed algorithms. ResultsThe theorems for computing the chromatic polynomial and chromatic index polynomial are thoroughly demonstrated and validated in this article. Additionally, the algorithms used in these computations are introduced and explained.

The Euler Characteristic of Parabolic Sheaves

Objectives: The primary aim of this study is to explicitly determine the Euler characteristic of the parabolic sheaves with rank 2 on a smooth projective algebraic surface defined over complex numbers with the smooth irreducible parabolic divisor . Methods: The computation of the parabolic Hilbert polynomial involves the use of -filtered sheaves on a smooth projective surface , with weights corresponding to the points where the filtration jumps. The Riemann-Roch theorem and Chern class computation have also been used. Findings: The study provides explicit computations of the parabolic Hilbert polynomial as well as the parabolic Chern classes for parabolic rank 2 bundles. Novelty: This work contributes to the understanding of parabolic sheaves on smooth projective surfaces, bridging the gap between different constructions of stable bundles. The explicit computation of the parabolic Hilbert polynomial for rank 2 bundles adds valuable insights to the study of moduli spaces of parabolic bundles. Keywords: Euler characteristic, Hilbert polynomial, Chern class, Parabolic sheaves, Smooth projective algebraic surface

How to Make Rational Arguments Practical and Extractable

We investigate proof systems where security holds against rational parties instead of malicious ones. Our starting point is the notion of rational arguments, a variant of rational proofs (Azar and Micali, STOC 2012) where security holds against rational adversaries that are also computationally bounded. Rational arguments are an interesting primitive because they generally allow for very efficient protocols, and in particular sublinear verification (i.e. where the Verifier does not have to read the entire input). In this paper we aim at narrowing the gap between literature on rational schemes and real world applications. Our contribution is two-fold. We provide the first construction of rational arguments for the class of polynomial computations that is practical (i.e., it can be applied to real-world computations on reasonably common hardware) and with logarithmic communication. Techniques-wise, we obtain this result through a compiler from information-theoretic protocols and rational proofs for polynomial evaluation. The latter could be of independent interest. As a second contribution, we propose a new notion of extractability for rational arguments. Through this notion we can obtain arguments where knowledge of a witness is incentivized (rather than incentivizing mere soundness). We show how our aforementioned compiler can also be applied to obtain efficient extractable rational arguments for N P .

On topological analysis of two-dimensional covalent organic frameworks via M-polynomial

Covalent organic frameworks (ZnP-COFs) made of zinc-porphyrin have become effective materials with a variety of uses, including gas storage and catalysis. To simulate the structural and electrical features of ZnP-COFs, this study goes into the computation of polynomials utilizing degree-based indices. We gave a methodical study of these polynomial computations using Excel, illustrating the complex interrelationships between the various indices. Degree-based indices provide valuable insights into the connectivity of vertices within a network. M-polynomials, on the other hand, offer a mathematical framework for representing and studying the properties of 2D COFs. By encoding structural information into a polynomial form, M-polynomials facilitate the calculation of various topological indices, including the Wiener index, Zagreb indices, and more. The different behavior of ZnP-COFs based on degree-based indices was illustrated graphically, and this comparison provided insightful information for prospective applications and the construction of innovative ZnP-COF structures. Moreover, we discuss the relevance of these techniques in the broader context of materials science and the design of functional covalent organic frameworks.

Efficient ASIC Architecture for Low Latency Classic McEliece Decoding

Post-quantum cryptography addresses the increasing threat that quantum computing poses to modern communication systems. Among the available “quantum-resistant” systems, the Classic McEliece key encapsulation mechanism (KEM) is positioned as a conservative choice with strong security guarantees. Building upon the code-based Niederreiter cryptosystem, this KEM enables high performance encapsulation and decapsulation and is thus ideally suited for applications such as the acceleration of server workloads. However, until now, no ASIC architecture is available for low latency computation of Classic McEliece operations. Therefore, the present work targets the design, implementation and optimization of a tailored ASIC architecture for low latency Classic McEliece decoding. An efficient ASIC design is proposed, which was implemented and manufactured in a 22 nm FDSOI CMOS technology node. We also introduce a novel inversionless architecture for the computation of error-locator polynomials as well as a systolic array for combined syndrome computation and polynomial evaluation. With these approaches, the associated optimized architecture improves the latency of computing error-locator polynomials by 47% and the overall decoding latency by 27% compared to a state-of-the-art reference, while requiring only 25% of the area.

Quantum solution of the relationship between the 19-vertex model and the Jones polynomial

The challenge is to create an efficient quantum algorithm for the bosonic model capable of calculating the Jones polynomials for a knot resulting from interweaving or interlacing n-vertices. This weave is the construction of braid group representations from nineteen-vertex model. We present eigenbases and eigenvalues for lattice generators and their usefulness for the direct computation of Jones polynomials. The calculation shows that the Temperley-Lieb operators can be used for any braid word. Therefore, we propose a quantum sequence using these singular operators as quantum gates operating on the state of n qubits. We show that quantum calculations give the Jones polynomial for achiral knots and links.

Addition and intersection of linear time-invariant behaviors

We define and analyze the operations of addition and intersection of linear time-invariant systems in the behavioral setting, where systems are viewed as sets of trajectories rather than input–output maps. The classical definition of addition of input–output systems is addition of the outputs with the inputs being equal. In the behavioral setting, addition of systems is defined as addition of all variables. Intersection of linear time-invariant systems was considered before only for the autonomous case in the context of “common dynamics” estimation. We generalize the notion of common dynamics to open systems (systems with inputs) as intersection of behaviors. This is done by proposing trajectory-based definitions. The main results of the paper are (1) characterization of the link between the complexities (number of inputs and order) of the sum and intersection systems, (2) algorithms for computing their kernel and image representations and (3) a duality property of the two operations. Our approach combines polynomial and numerical linear algebra computations.

Energy-Efficient and Low-Complexity Transmission Control With SWIPT-NOMA for Green Cellular Networks

In this study, we consider an energy-efficient and low-complexity transmission control in a SWIPT-NOMA-based green cellular network (GCN) that consists of a green base station (GBS) and green users (GUEs). First, we formulate a non-convex problem that minimizes transmit power consumption while supporting minimum downlink user service rate, downlink data queue stability, and user battery availability. Then, we transform the problem into a Lyapunov-drift-penalty minimization problem, which can determine a new resource allocation scheme that balances transmit power consumption and queue stability. Second, the Lyapunov-drift-penalty problem is decomposed into subchannel assignment, power allocation, and power splitting (PS) ratio control problems. The subchannel assignment problem is solved using a matching theory-based low-complexity algorithm. The power allocation and PS ratio control problems are solved using the alternating optimization (AO) approach and bisection method. This decomposed subproblem-based control also enables distributed control between the GBS and GUEs. Third, we prove the convergence, optimality, and polynomial computation complexity of the proposed algorithm. Lastly, we demonstrate that the proposed control outperforms the benchmark controls regarding transmit power consumption and the achievable rate. Owing to the optimality and low complexity, the proposed control can be efficiently applied to large-scale and distributed GCNs in sixth-generation environments.

Coordinated seru scheduling and distribution operation problems with DeJong’s learning effects

Inspired by the advantage of just-in-time philosophy, zero-inventory is common in many industries, especially those need fast response with a short lifespan. Accordingly, this paper focuses on coordinated production scheduling and distribution operation problems considering workers’ DeJong’s learning effects in seru production system (SPS), in which seru is a relatively new-type manufacturing mode originating from Japan and can achieve fast response in practice. Two variants of coordinated seru scheduling and distribution operation problems are studied, and the corresponding 0–1 integer programming model is formulated. By analyzing the mathematical property, the polynomial computation time of the former is able to be determined, and an intractability and NP-hardness proof is provided for the latter. The dynamic programming-based exact algorithm and the heuristic ant colony optimization algorithm are developed respectively. Computational experiments are conducted finally, and a series of experimental results indicate that the Dejong’s learning effect has a significant influence on coordinated seru scheduling and distribution operation problems, meanwhile a remarkable benefit (the average improvement is 16.97%) can be achieved by the coordinated production scheduling and distribution operation in SPS.

Polynomial-Based Approach for Efficient Function Computation of Symmetric Matrices Using Restarted Heavy Ball Method

This paper presents a polynomial-based approach for efficiently computing functions ofsymmetric matrices by leveraging the Restarted Heavy Ball (RHB) method. The RHB method is employedto overcome the slow convergence issue commonly encountered when computing functions of symmetricmatrices. The key idea of our approach is to approximate the desired function using a polynomial. Byrepresenting the function as a polynomial, we can leverage the efficient computation of polynomials toaccelerate the overall function computation process. We introduce a systematic methodology forconstructing an optimal polynomial approximation that minimizes the approximation error. To furtherenhance the convergence speed, we incorporate the Restarted Heavy Ball method into our polynomialbased approach. The Restarted Heavy Ball iteration is applied after a certain number of iterations to resetthe computation process and mitigate the slow convergence behavior. The experimental results and analysisvalidate the effectiveness and practicality of our approach, highlighting its potential for various applicationsinvolving function computations of symmetric matrices. Overall, our polynomial-based approach,integrated with the Restarted Heavy Ball method, offers an efficient and accurate solution for computingfunctions of symmetric matrices. The experimental results and analysis validate the effectiveness andpracticality of our approach, highlighting its potential for various applications involving functioncomputations of symmetric matrices.

Модальный синтез регуляторов полиномиальным методом для систем автоматического управления с избыточной размерностью вектора управления

When using polynomial methods of synthesis of multichannel regulators, there is a need for polynomial matrix calculus. However, when using this method, objects with the number of output channels equal to the number of input channels are mainly considered. This is necessary for the convenience of solving a system of linear algebraic equations in matrix polynomial computation. At the same time, a fairly large number of real technical systems have an unequal number of input and output channels. At the same time, the issue of the synthesis of generators by the polynomial method for multichannel objects with an unequal number of input and output impacts has not been worked out sufficiently deeply. One of the special cases of such systems can be considered systems with excessive dimension of the control vector. Within the framework of this work, examples of such systems and the purposes of their use are given. An illustrative example of a linear model of an unstable object with three channels for input action and two channels for output action is given. It is necessary to achieve certain quality indicators of the output vector quantity, while the control is carried out in the feedback of the system and is summed up with the input effect. The simplicity of the system under consideration is connected with the convenience of demonstrating on this example a modal synthesis method using a polynomial matrix decomposition of the transfer functions of an object and a controller for such a class of objects. In accordance with the recommendations presented in the algorithm for synthesizing regulators for objects with a non-square matrix transfer function, to solve the problem of synthesizing a regulator for systems with an excessive dimension of the control vector, the transfer function of the control object is represented as a right polynomial matrix decomposition, and the regulator is represented as a left one. During the demonstration of the example of the algorithm, some clarifications and edits are proposed in it.

Level-Based Blocking for Sparse Matrices: Sparse Matrix-Power-Vector Multiplication

The multiplication of a sparse matrix with a dense vector (SpMV) is a key component in many numerical schemes and its performance is known to be severely limited by main memory access. Several numerical schemes require the multiplication of a sparse matrix polynomial with a dense vector which is typically implemented as a sequence of SpMVs. This results in low performance and ignores the potential to increase the arithmetic intensity by reusing the matrix data from cache. In this work we use the recursive algebraic coloring engine (RACE) to enable blocking of sparse matrix data across the polynomial computations. In the graph representing the sparse matrix we form levels using a breadth-first search. Locality relations of these levels are then used to improve spatial and temporal locality when accessing the matrix data and to implement an efficient multithreaded parallelization. Our approach is independent of the matrix structure and avoids shortcomings of existing “blocking” strategies in terms of hardware efficiency and parallelization overhead. We quantify the quality of our implementation using performance modelling and demonstrate speedups of up to 3× and 5× compared to an optimal SpMV-based baseline on a single multicore chip of recent Intel and AMD architectures. Various numerical schemes like <inline-formula><tex-math notation="LaTeX">$s$</tex-math></inline-formula> -step Krylov solvers, polynomial preconditioners and power clustering algorithms will benefit from our development.

A Reconfigurable Hardware Architecture for Miscellaneous Floating-Point Transcendental Functions

Transcendental functions are an important part of algorithms in many fields. However, the hardware accelerators available today for transcendental functions typically only support one such function. Hardware accelerators that can support miscellaneous transcendent functions are a waste of hardware resources. In order to solve these problems, this paper proposes a reconfigurable hardware architecture for miscellaneous floating-point transcendental functions. The hardware architecture supports a variety of transcendental functions, including floating-point sine, cosine, arctangent, exponential and logarithmic functions. It adopts the method of a lookup table combined with a polynomial computation and reconfigurable technology to achieve the accuracy of two units of least precision (ulp) with 3.75 KB lookup tables and one core computing module. In addition, the hardware architecture uses retiming technology to realize the different operation times of each function. Experiments show that the hardware accelerators proposed can operate at a maximum frequency of 220 MHz. The full-load power consumption and areas are only 0.923 mW and 1.40×104μm2, which are reduced by 47.99% and 38.91%, respectively, compared with five separate superfunction hardware accelerators.

Multi-Party Secure Computation of Multi-Variable Polynomials

Multi-Party Secure Computation of Multi-Variable Polynomials

A computational approach to analyze the Hadamard quasigroup product

&lt;abstract&gt;&lt;p&gt;Based on the binary product described by any Latin square, the Hadamard quasigroup product is introduced in this paper as a natural generalization of the classical Hadamard product of matrices. The successive iteration of this new product is endowed with a cyclic behaviour that enables one to define a pair of new isomorphism invariants of Latin squares. Of particular interest is the set of Latin squares for which this iteration preserves the Latin square property, which requires the existence of successive localized Latin transversals within the Latin square under consideration. In order to enumerate and classify, up to isomorphism, these Latin squares, we propose a computational algebraic geometry approach based on the computation of reduced Gröbner bases. To illustrate this point, we obtain the classification of the sought Latin squares, for order up to six, by using the open computer algebra system for polynomial computations Singular.&lt;/p&gt;&lt;/abstract&gt;