104.41 A property of the Lagrange interpolation polynomial

2020 ◽  
Vol 104 (561) ◽  
pp. 548-551
Author(s):  
Mowaffaq Hajja
Keyword(s):  
Lagrange Interpolation ◽  
Interpolation Polynomial ◽  
Lagrange Interpolation Polynomial

scholarly journals Fuzzy Volterra Integro-Differential Equations Using General Linear Method

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 381 ◽  
Author(s):  
Zanariah Abdul Majid ◽  
Faranak Rabiei ◽  
Fatin Abd Hamid ◽  
Fudziah Ismail
Keyword(s):  
Differential Equations ◽  
Lagrange Interpolation ◽  
Linear Method ◽  
Interpolation Polynomial ◽  
General Linear ◽  
Kutta Method ◽  
Lagrange Interpolation Polynomial ◽  
Runge Kutta Method ◽  
General Linear Method ◽  
Generalized Hukuhara Differentiability

In this paper, a fuzzy general linear method of order three for solving fuzzy Volterra integro-differential equations of second kind is proposed. The general linear method is operated using the both internal stages of Runge-Kutta method and multivalues of a multisteps method. The derivation of general linear method is based on the theory of B-series and rooted trees. Here, the fuzzy general linear method using the approach of generalized Hukuhara differentiability and combination of composite Simpson’s rules together with Lagrange interpolation polynomial is constructed for numerical solution of fuzzy volterra integro-differential equations. To illustrate the performance of the method, the numerical results are compared with some existing numerical methods.

scholarly journals On Secret Sharing with Newton’s Polynomial for Multi-Factor Authentication

Cryptography ◽  
2020 ◽  
Vol 4 (4) ◽  
pp. 34
Author(s):  
Sergey Bezzateev ◽  
Vadim Davydov ◽  
Aleksandr Ometov
Keyword(s):  
Access Control ◽  
Secret Sharing ◽  
Lagrange Interpolation ◽  
Interpolation Formula ◽  
Building Blocks ◽  
Interpolation Polynomial ◽  
Lagrange Interpolation Polynomial ◽  
Crucial Component

Security and access control aspects are becoming more and more essential to consider during the design of various systems and the tremendous growth of digitization. One of the related key building blocks in this regard is, essentially, the authentication process. Conventional schemes based on one or two authenticating factors can no longer provide the required levels of flexibility and pro-activity of the access procedures, thus, the concept of threshold-based multi-factor authentication (MFA) was introduced, in which some of the factors may be missing, but the access can still be granted. In turn, secret sharing is a crucial component of the MFA systems, with Shamir’s schema being the most widely known one historically and based on Lagrange interpolation polynomial. Interestingly, the older Newtonian approach to the same problem is almost left without attention. At the same time, it means that the coefficients of the existing secret polynomial do not need to be re-calculated while adding a new factor. Therefore, this paper investigates this known property of Newton’s interpolation formula, illustrating that, in specific MFA cases, the whole system may become more flexible and scalable, which is essential for future authentication systems.

Necessary and Sufficient Conditions for Mean Convergence of Lagrange Interpolation for Erdős Weights II

1996 ◽  
Vol 48 (4) ◽  
pp. 737-757 ◽  
Author(s):  
S. B. Damelin ◽  
D. S. Lubinsky
Keyword(s):  
Continuous Function ◽  
Lagrange Interpolation ◽  
Sufficient Conditions ◽  
Interpolation Polynomial ◽  
Weighting Factor ◽  
Lagrange Interpolation Polynomial ◽  
Mean Convergence ◽  
Turán Theorem ◽  
Image Position ◽  
Necessary And Sufficient

AbstractWe complete our investigations of mean convergence of Lagrange interpolation at the zeros of orthogonal polynomials pn(W2, x) for Erdős weights W2 = e-2Q. The archetypal example is Wk,α = exp(—Qk,α), whereα > 1, k ≥ 1, and is the k-th iterated exponential. Following is our main result: Let 1 < p < 4 and α ∊ ℝ Let Ln[f] denote the Lagrange interpolation polynomial to ƒ at the zeros of pn(W2, x) = pn(e-2Q, x). Then forto hold for every continuous function ƒ:ℝ. —> ℝ satisfyingit is necessary and sufficient that α > 1/p. This is, essentially, an extension of the Erdös-Turan theorem on L2 convergence. In an earlier paper, we analyzed convergence for all p > 1, showing the necessity and sufficiency of using the weighting factor 1 + Q for all p > 4. Our proofs of convergence are based on converse quadrature sum estimates, that are established using methods of H. König.

scholarly journals A WSN Layer-Cluster Key Management Scheme Based on Quadratic Polynomial and Lagrange Interpolation Polynomial

Sensors ◽  
2020 ◽  
Vol 20 (16) ◽  
pp. 4388
Author(s):  
Xiaogang Wang ◽  
Zhongfan Yang ◽  
Zhiqiang Feng ◽  
Jun Zhao
Keyword(s):  
Key Management ◽  
Lagrange Interpolation ◽  
Interpolation Polynomial ◽  
Identity Authentication ◽  
Quadratic Polynomial ◽  
Lagrange Interpolation Polynomial ◽  
Session Key ◽  
Group Key ◽  
Management Scheme ◽  
Key Management Scheme

Since current key management schemes are mainly designed for static and planar networks, they are not very suitable for the layer-cluster wireless sensor networks (WSNs), a WSN layer-cluster key management scheme based on quadratic polynomial and Lagrange interpolation polynomial is proposed, in which the main idea of this scheme along the research line of broadcast identity authentication, session key, group key, network key and personal key. Specifically, authentication key can be established on the basis of Fourier series for identity authentication; session key is established by a multiple asymmetric quadratic polynomial, in which session key information is encrypted by the authentication key to ensure the security of intermediate interactive information; based on the former two keys, group key is established on the basis of Lagrange interpolation polynomial, in which the nodes of the cluster are not directly involved; the generation and management of network key is similar to the group key, in which the establishment idea is to regard the BS and all cluster heads as a group; the generation and management of personal key is also similar to the group key, the difference is that the personal key can be obtained by cluster nodes through getting the Lagrange interpolation polynomial coefficients based on their own random key information. It is analyzed that the proposed layer-cluster key management scheme can guarantee the identity of network nodes firstly through forward authentication and reverse authentication, and session key, group key and network key will guarantee the independence of the keys’ management and avoids the problem of single point failure compared with LEAP protocol, and personal key will guarantee the privacy of network.

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