Polynomial Interpolation Techniques Research Articles

Game on: a performance comparison of interpolation techniques applied to Shamir’s secret sharing

Abstract Public-key encryption is typically managed through a public key infrastructure. However, it relies on a central control point, the certification authority, which acts as a single point of failure. Recent technological advancements have led to the need for decentralized cryptographic protocols. This paper presents a comprehensive study on enhancing public-key encryption via threshold cryptography and multiparty computation to ensure robust security in decentralized systems. The focus lies in exploring various polynomial interpolation techniques within Shamir’s secret sharing scheme, particularly addressing the efficiency and practicality of Newton interpolation, fast Fourier transformation (FFT), and advanced versions of Lagrange’s method. Utilizing SageMath for a dedicated testing environment, the research investigates the swiftest interpolation methods for secret recovery, introducing new shares into the system, and evaluating the impact of optimizations on performance. The findings highlight FFT as the most effective interpolation method in speed and efficiency, albeit with limitations on the number of shares that can be processed. This paper critically evaluates these interpolation techniques against practical constraints and aims to answer pivotal research questions regarding the optimal approach for large-scale scenarios, challenging existing notions on the efficiency of Newton’s method and providing experimental evidence to support the superiority of FFT in specific contexts.

Improved Accuracy for Trotter Simulations Using Chebyshev Interpolation

Quantum metrology allows for measuring properties of a quantum system at the optimal Heisenberg limit. However, when the relevant quantum states are prepared using digital Hamiltonian simulation, the accrued algorithmic errors will cause deviations from this fundamental limit. In this work, we show how algorithmic errors due to Trotterized time evolution can be mitigated through the use of standard polynomial interpolation techniques. Our approach is to extrapolate to zero Trotter step size, akin to zero-noise extrapolation techniques for mitigating hardware errors. We perform a rigorous error analysis of the interpolation approach for estimating eigenvalues and time-evolved expectation values, and show that the Heisenberg limit is achieved up to polylogarithmic factors in the error. Our work suggests that accuracies approaching those of state-of-the-art simulation algorithms may be achieved using Trotter and classical resources alone for a number of relevant algorithmic tasks.

A method to extract macroscopic interface data from microscale rough/porous wall flow fields

Performing geometry-resolved simulations of flows over rough and porous walls is highly expensive due to their multiscale characteristics. Effective models that circumvent this difficulty are often used to investigate the interaction between the free-fluid and such complex walls. These models, by construction, employ an intrinsic averaging process and capture only macroscopic physical processes. However, physical experiments or direct simulations yield micro- and macroscale information, and isolating the macroscopic effect from them is crucial for rigorously validating the accuracy of effective models. Despite the increasing use of effective models, this aspect received the least attention in the literature. This paper presents an efficient averaging technique to extract macroscopic interface data from the flow field obtained via direct simulations or physical experiments. The proposed methodology employs a combination of signal processing and polynomial interpolation techniques to capture the macroscopic information. Results from the ensemble averaging are used as the reference to quantify the accuracy of the proposed method. Compared to the ensemble averaging, the proposed method, while retaining accuracy, is cost-effective for rough and porous walls. To the best of our knowledge, this is the only averaging method that works for poroelastic walls, for which the ensemble averaging fails. Moreover, it applies equally to viscous- and inertia-dominated flows over irregular surfaces.

A class of projected-search methods for bound-constrained optimization

Projected-search methods for bound-constrained optimization are based on performing a search along a piecewise-linear continuous path obtained by projecting a search direction onto the feasible region. A potential benefit of a projected-search method is that many changes to the active set can be made at the cost of computing a single search direction. As the objective function is not differentiable along the search path, it is not possible to use a projected-search method with a step that satisfies the Wolfe conditions, which require the directional derivative of the objective function at a point on the path. For this reason, methods based in full or in part on a simple backtracking procedure must be used to give a step that satisfies an ‘Armijo-like’ sufficient decrease condition. As a consequence, conventional projected-search methods are unable to exploit sophisticated safeguarded polynomial interpolation techniques that have been shown to be effective for the unconstrained case. This paper describes a new framework for the development of a general class of projected-search methods for bound-constrained optimization. At each iteration, a descent direction is computed with respect to a certain extended active set. This direction is used to specify a search direction that is used in conjunction with a step length computed by a quasi-Wolfe search. The quasi-Wolfe search is designed specifically for use with a piecewise-linear search path and is similar to a conventional Wolfe line search, except that a step is accepted under a wider range of conditions. These conditions take into consideration steps at which the restriction of the objective function on the search path is not differentiable. Standard existence and convergence results associated with a conventional Wolfe line search are extended to the quasi-Wolfe case. In addition, it is shown that under a standard nondegeneracy assumption, any method within the framework will identify the optimal active set in a finite number of iterations. Computational results are given for a specific projected-search method that uses a limited-memory quasi-Newton approximation of the Hessian. The results show that, in this context, a quasi-Wolfe search is substantially more efficient and reliable than an Armijo-like search based on simple backtracking. Comparisons with a state-of-the-art bound-constrained optimization package are also presented.

Authentication and encryption protocol with revocation and reputation management for enhancing 5G-V2X security

This paper presents a novel robust solution for key management, message encryption, and authentication, offering enhanced security for 5G-V2X communication. By incorporating bilinear pairing, and AES encryption, the protocol achieves a high level of security. The protocol consists of two key phases: key management and distribution, and message authentication and encryption. In the key management phase, secret sharing and polynomial interpolation techniques are used to generate and distribute the master secret key. The message authentication and encryption utilize bilinear pairing for integrity and origin authentication and symmetric encryption for confidentiality. To address the real-time update limitations of traditional revoking approaches, a reputation-based vehicle revocation mechanism is proposed. This mechanism allows fair revoking based on reputation scores and authenticated feedback, ensuring effective resource management. The proposed protocol utilizes the MIRACL library to implement the bilinear pairing operations. Comparative analysis demonstrates the superior performance of the protocol compared to other bilinear pairing protocols. The results show that the proposed protocol enhances authentication and encryption while incurring short times for message signing and verification (1.925 ms and 7.115 ms, respectively). Furthermore, the protocol supports signature aggregation using bilinear properties, resulting in constant verification time regardless of the number of signatures.

Localized Chebyshev and MLS collocation methods for solving 2D steady state nonlocal diffusion and peridynamic equations

The localized Chebyshev collocation method and MLS collocation method are presented to obtain the solution of two-dimensional nonlocal diffusion and peridynamic equations. The Chebyshev polynomial and MLS interpolation techniques are used to construct shape functions in the frame of the collocation method. Low computational cost and high accuracy are the main advantages of these two methods for solving nonlocal diffusion and peridynamic equations. Several numerical examples are provided to show the validity and applicability of the proposed method with the regular and irregular domains. Numerical experiments indicate that the localized Chebyshev collocation method has high accuracy for nonlocal problems with continuous solutions. The MLS collocation method is more efficient and can maintain good behavior for problems with discontinuous solutions.

Fingerprint Database Enhancement by Applying Interpolation and Regression Techniques for IoT-based Indoor Localization

Most applied indoor localization is based on distance and fingerprint techniques. The distance-based technique converts specific parameters to a distance, while the fingerprint technique stores parameters as the fingerprint database. The widely used Internet of Things (IoT) technologies, e.g., Wi-Fi and ZigBee, provide the localization parameters, i.e., received signal strength indicator (RSSI). The fingerprint technique advantages over the distance-based method as it straightforwardly uses the parameter and has better accuracy. However, the burden in database reconstruction in terms of complexity and cost is the disadvantage of this technique. Some solutions, i.e., interpolation, image-based method, machine learning (ML)-based, have been proposed to enhance the fingerprint methods. The limitations are complex and evaluated only in a single environment or simulation. This paper proposes applying classical interpolation and regression to create the synthetic fingerprint database using only a relatively sparse RSSI dataset. We use bilinear and polynomial interpolation and polynomial regression techniques to create the synthetic database and apply our methods to the 2D and 3D environments. We obtain an accuracy improvement of 0.2m for 2D and 0.13m for 3D by applying the synthetic database. Adding the synthetic database can tackle the sparsity issues, and the offline fingerprint database construction will be less burden. Doi: 10.28991/esj-2021-SP1-012 Full Text: PDF

Временные корреляционные методы оценки задержки комплексных сигналов, некратной периоду дискретизации, на основе полиномиальной интерполяции.

This paper deals with the problem of subsample time delay estimation of complex signal based on polynomial interpolation. Time delay estimation is performed by cross-correlation time approach. Three polynomial interpolation techniques applied to the discrete complex cross-correlation function in the neighborhood of its maximum are proposed. These methods show high processing speed and allow obtaining accurate real-valued time delay estimation when digital complex signals are processed. The comparative analysis between these methods is performed. A rigorous analytical solution for the correction of time delay estimation for one of the proposed methods is obtained for the case of the third-order polynomial interpolation. This solution is applied for an equidistant grid of discrete cross-correlation function samples. One can improve the accuracy of time delay estimates by using aforementioned correction. A numerical simulation is performed to quantify the accuracy of the time delay estimates when using the proposed methods for the case where a stationary random process described by the first-order autoregressive mode is chosen as a model of original signal. The main results were presented and discussed at XIV All-Russian conference “Radar and telecommunication”.

Optimal estimation of Gaussian mixtures via denoised method of moments

The method of moments (Philos. Trans. R. Soc. Lond. Ser. A 185 (1894) 71–110) is one of the most widely used methods in statistics for parameter estimation, by means of solving the system of equations that match the population and estimated moments. However, in practice and especially for the important case of mixture models, one frequently needs to contend with the difficulties of non-existence or nonuniqueness of statistically meaningful solutions, as well as the high computational cost of solving large polynomial systems. Moreover, theoretical analyses of the method of moments are mainly confined to asymptotic normality style of results established under strong assumptions. This paper considers estimating a $k$-component Gaussian location mixture with a common (possibly unknown) variance parameter. To overcome the aforementioned theoretic and algorithmic hurdles, a crucial step is to denoise the moment estimates by projecting to the truncated moment space (via semidefinite programming) before solving the method of moments equations. Not only does this regularization ensure existence and uniqueness of solutions, it also yields fast solvers by means of Gauss quadrature. Furthermore, by proving new moment comparison theorems in the Wasserstein distance via polynomial interpolation and majorization techniques, we establish the statistical guarantees and adaptive optimality of the proposed procedure, as well as oracle inequality in misspecified models. These results can also be viewed as provable algorithms for generalized method of moments (Econometrica 50 (1982) 1029–1054) which involves nonconvex optimization and lacks theoretical guarantees.

Superresolution without separation

Abstract This article provides a theoretical analysis of diffraction-limited superresolution, demonstrating that arbitrarily close point sources can be resolved in ideal situations. Precisely, we assume that the incoming signal is a linear combination of $M$ shifted copies of a known waveform with unknown shifts and amplitudes, and one only observes a finite collection of evaluations of this signal. We characterize properties of the base waveform such that the exact translations and amplitudes can be recovered from $2M+1$ observations. This recovery can be achieved by solving a weighted version of basis pursuit over a continuous dictionary. Our analysis shows that $\ell_1$-based methods enjoy the same separation-free recovery guarantees as polynomial root finding techniques, such as de Prony’s method or Vetterli’s method for signals of finite rate of innovation. Our proof techniques combine classical polynomial interpolation techniques with contemporary tools from compressed sensing.

Age Estimation of Polymeric Insulation in Absorption of Chemical Components

The investigation of the lifetime execution of polymeric insulator is frequently in light of deficient information. Specialists confront difficulties of not having long perception time to accumulate finish lifetime information so as to make forecast with conviction. In this paper, we are proposing to utilize the best fitting polynomial interpolation method to deal with decide the lifetime show and extrapolate data towards the period of polymeric insulators. To assemble the data, for example, the change of convergence of substance component of polymeric insulation because of various anxiety conditions, high temperature vulcanized (HTV) silicone rubberexamples were subjected to quickened debasement utilizing an Accelerated Weathering Tester (QUV). Every example was subjected to quickened maturing cycle in agreement to ASTM G154 cycle 4 for an aggregate of 3000 hours. The maturing cycle experiences a cycle of 70°C and bright introduction at light power of 1.55 W/m2 for eight hours taken after by four hour buildup at 50°C. Chemical component focuses, for example, Carbon, Oxygen, Silicon and Aluminum were acquired and assessed utilizing a Scanning Electron Microscope (SEM) with energy dispersive xray (EDX). The convergences of chemical components are broke down utilizing polynomial interpolation strategy for different requests. In the first place request to fifth request polynomial interpolation techniques are examined to decide the best fitting bend without essentialness difference from the genuine esteem or genuine esteem. The interpolation curve is utilized to gauge the age of an obscure matured specimen. This examination has uncovered a novel technique for deciding a model that could portray the lifetime of polymeric insulators and extrapolation strategy for evaluating the age for polymer insulator.

A FRACTIONAL DELAY FIR FILTER BASED ON LAGRANGE INTERPOLATION OF FARROW STRUCTURE

An efficient implementation technique for the Lagrange interpolation is derived. This formulation called the Farrow structure leads to a version of Lagrange interpolation that is well suited to time varying FD filtering. Lagrange interpolation is mostly used for fractional delay approximation as it can be used for increasing the sampling rate of signals and systems. Lagrange interpolation is one of the representatives for a class of polynomial interpolation techniques. The computational cost of this structure is reduced as the number of multiplications are minimised in the new structure when compared with the conventional structure.

Accurate estimation of solvation free energy using polynomial fitting techniques.

This report details an approach to improve the accuracy of free energy difference estimates using thermodynamic integration data (slope of the free energy with respect to the switching variable λ) and its application to calculating solvation free energy. The central idea is to utilize polynomial fitting schemes to approximate the thermodynamic integration data to improve the accuracy of the free energy difference estimates. Previously, we introduced the use of polynomial regression technique to fit thermodynamic integration data (Shyu and Ytreberg, J Comput Chem, 2009, 30, 2297). In this report we introduce polynomial and spline interpolation techniques. Two systems with analytically solvable relative free energies are used to test the accuracy of the interpolation approach. We also use both interpolation and regression methods to determine a small molecule solvation free energy. Our simulations show that, using such polynomial techniques and nonequidistant λ values, the solvation free energy can be estimated with high accuracy without using soft-core scaling and separate simulations for Lennard-Jones and partial charges. The results from our study suggest that these polynomial techniques, especially with use of nonequidistant λ values, improve the accuracy for ΔF estimates without demanding additional simulations. We also provide general guidelines for use of polynomial fitting to estimate free energy. To allow researchers to immediately utilize these methods, free software and documentation is provided via http://www.phys.uidaho.edu/ytreberg/software.

A Graph-Based Unified Technique for Computing and Representing Coefficients over Finite Fields

This paper presents the generalized theory and an efficient graph-based technique for the calculation and representation of coefficients of multivariate canonic polynomials over arbitrary finite fields in any polarity. The technique presented for computing coefficients is unlike polynomial interpolation or matrix-based techniques and takes into consideration efficient graph-based forms which can be available as an existing resource during synthesis, verification, or simulation of digital systems. Techniques for optimization of the graph-based forms for representing the coefficients are also presented. The efficiency of the algorithm increases for larger fields. As a test case, the proposed technique has been applied to benchmark circuits over GF(2m). The experimental results show that the proposed technique can significantly speed up execution time.

The complexity of counting colourings and independent sets in sparse graphs and hypergraphs

We consider certain counting problems involving colourings of graphs and independent sets in hypergraphs. Using polynomial interpolation techniques, we show that these problems are #P-complete. Therefore, efficient approximate counting is the most one can realistically expect to achieve. Rapidly mixing Markov chains which can be used for approximately solving some of these counting problems have been recently developed by the author and others.

Three-dimensional projection of curvilinear geological features through direction cosine interpolation of structural field observations

A UNIX program in AWK is presented for densifying and projecting distributed 3D data points that occur on a common geological structural surface. Polynomial and hybrid B-spline interpolation techniques are optimized for both sparse and abundant regional, clustered structural data sets. This tool, named push.awk, will be most useful in geological situations in which structures are constrained by several geometrical observations, are parts of relatively continuous irregular curviplanar surfaces and whose behavior can be predicted approximately linearly with depth at some point along the structure. Suggested applications are the determination of some types of variably plunging folds in mine exploration, depth prediction of regionally continuous shear zones and brittle faults, development of 3D structural fabric trajectories, horizon propagation, and plutonic boundary geometry evaluation. Simple uniaxial and variable projection methods need to be used in an appropriate context and in conjunction with the entire range of available statistical and interpretive 3D tools in order to achieve high-quality geological visualizations. It is the effective incorporation of geometric field data into visualization and predictive geotechnologies that will increase the potential for subsurface interpretation in complex geological settings.

Reconstructing Digital Signals Using Shannon's Sampling Theorem

Researchers must be cognizant of the frequency content of analog signals that they are collecting. Knowing the frequency content allows the researcher to determine the minimum sampling frequency of the data (Nyquist critical frequency), ensuring that the digital data will have all of the frequency characteristics of the original signal. The Nyquist critical frequency is 2 times greater than the highest frequency in the signal. When sampled at a rate above the Nyquist, the digital data will contain all of the frequency characteristics of the original signal but may not present a correct time-series representation of the signal. In this paper, an algorithm known as Shannon's Sampling Theorem is presented that correctly reconstructs the time-series profile of any signal sampled above the Nyquist critical frequency. This method is superior to polynomial or spline interpolation techniques in that it can reconstruct peak values found in the original signal but missing from the sampled data time-series.