**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**69

# Search results for: p-y curves

##### 69 The Elliptic Curves y2 = x3 - t2x over Fp

**Authors:**
Ahmet Tekcan

**Abstract:**

Let p be a prime number, Fp be a finite field and t ∈ F*p= Fp- {0}. In this paper we obtain some properties of ellipticcurves Ep,t: y2= y2= x3- t2x over Fp. In the first sectionwe give some notations and preliminaries from elliptic curves. In the second section we consider the rational points (x, y) on Ep,t. Wegive a formula for the number of rational points on Ep,t over Fnp for an integer n ≥ 1. We also give some formulas for the sum of x?andy?coordinates of the points (x, y) on Ep,t. In the third section weconsider the rank of Et: y2= x3- t2x and its 2-isogenous curve Et over Q. We proved that the rank of Etand Etis 2 over Q. In the last section we obtain some formulas for the sums Σt∈F?panp,t for an integer n ≥ 1, where ap,t denote the trace of Frobenius.

**Keywords:**
Elliptic curves over finite fields,
rational points onelliptic curves,
rank,
trace of Frobenius.

##### 68 Some Characterizations of Isotropic Curves In the Euclidean Space

**Authors:**
Süha Yılmaz,
Melih Turgut

**Abstract:**

**Keywords:**
Classical Differential Geometry,
Euclidean space,
Minimal Curves,
Isotropic Curves,
Pseudo Helix.

##### 67 The Number of Rational Points on Elliptic Curves y2 = x3 + b2 Over Finite Fields

**Authors:**
Betül Gezer,
Hacer Özden,
Ahmet Tekcan,
Osman Bizim

**Abstract:**

Let p be a prime number, Fpbe a finite field and let Qpdenote the set of quadratic residues in Fp. In the first section we givesome notations and preliminaries from elliptic curves. In the secondsection, we consider some properties of rational points on ellipticcurves Ep,b: y2= x3+ b2 over Fp, where b ∈ F*p. Recall that theorder of Ep,bover Fpis p + 1 if p ≡ 5(mod 6). We generalize thisresult to any field Fnp for an integer n≥ 2. Further we obtain someresults concerning the sum Σ[x]Ep,b(Fp) and Σ[y]Ep,b(Fp), thesum of x- and y- coordinates of all points (x, y) on Ep,b, and alsothe the sum Σ(x,0)Ep,b(Fp), the sum of points (x, 0) on Ep,b.

**Keywords:**
Elliptic curves over finite fields,
rational points on elliptic curves.

##### 66 New DES based on Elliptic Curves

**Authors:**
Ghada Abdelmouez M.,
Fathy S. Helail,
Abdellatif A. Elkouny

**Abstract:**

**Keywords:**
DES,
Elliptic Curves,
hybrid system,
symmetricencryption.

##### 65 The Number of Rational Points on Elliptic Curves and Circles over Finite Fields

**Authors:**
Betül Gezer,
Ahmet Tekcan,
Osman Bizim

**Abstract:**

**Keywords:**
Elliptic curves over finite fields,
rational points on
elliptic curves and circles.

##### 64 On the Differential Geometry of the Curves in Minkowski Space-Time II

**Authors:**
Süha Yılmaz,
Emin Özyılmaz,
Melih Turgut

**Abstract:**

**Keywords:**
Frenet Apparatus,
Time-like Curves,
MinkowskiSpace-time.

##### 63 Analytical Development of a Failure Limit and Iso-Uplift Curves for Eccentrically Loaded Shallow Foundations

**Authors:**
N. Abbas,
S. Lagomarsino,
S. Cattari

**Abstract:**

Examining existing experimental results for shallow rigid foundations subjected to vertical centric load (N), accompanied or not with a bending moment (M), two main non-linear mechanisms governing the cyclic response of the soil-foundation system can be distinguished: foundation uplift and soil yielding. A soil-foundation failure limit, is defined as a domain of resistance in the two dimensional (2D) load space (N, M) inside of which lie all the admissible combinations of loads; these latter correspond to a pure elastic, non-linear elastic or plastic behavior of the soil-foundation system, while the points lying on the failure limit correspond to a combination of loads leading to a failure of the soil-foundation system. In this study, the proposed resistance domain is constructed analytically based on mechanics. Original elastic limit, uplift initiation limit and iso-uplift limits are constructed inside this domain. These limits give a prediction of the mechanisms activated for each combination of loads applied to the foundation. A comparison of the proposed failure limit with experimental tests existing in the literature shows interesting results. Also, the developed uplift initiation limit and iso-uplift curves are confronted with others already proposed in the literature and widely used due to the absence of other alternatives, and remarkable differences are noted, showing evident errors in the past proposals and relevant accuracy for those given in the present work.

**Keywords:**
Foundation uplift,
Iso-uplift curves,
Resistance
domain,
Soil yield.

##### 62 The Use of S Curves in Technology Forecasting and its Application On 3D TV Technology

**Authors:**
Gizem Intepe,
Tufan Koc

**Abstract:**

**Keywords:**
Patent analysis,
Technological forecasting. S curves,
3D TV

##### 61 Finding More Non-Supersingular Elliptic Curves for Pairing-Based Cryptosystems

**Authors:**
Pu Duan,
Shi Cui,
Choong Wah Chan

**Abstract:**

**Keywords:**
Family of group order,
kth root of unity,
non-supersingular elliptic curves polynomial field.

##### 60 Classification of the Bachet Elliptic Curves y2 = x3 + a3 in Fp, where p ≡ 1 (mod 6) is Prime

**Authors:**
Nazli Yildiz İkikardes,
Gokhan Soydan,
Musa Demirci,
Ismail Naci Cangul

**Abstract:**

**Keywords:**
Elliptic curves over finite fields,
quadratic residue,
cubic residue.

##### 59 Box Counting Dimension of the Union L of Trinomial Curves When α ≥ 1

**Authors:**
Kaoutar Lamrini Uahabi,
Mohamed Atounti

**Abstract:**

**Keywords:**
Feasible angles,
fractal dimension,
Minkowski
sausage,
trinomial curves,
trinomial equation.

##### 58 Circular Approximation by Trigonometric Bézier Curves

**Authors:**
Maria Hussin,
Malik Zawwar Hussain,
Mubashrah Saddiqa

**Abstract:**

We present a trigonometric scheme to approximate a circular arc with its two end points and two end tangents/unit tangents. A rational cubic trigonometric Bézier curve is constructed whose end control points are defined by the end points of the circular arc. Weight functions and the remaining control points of the cubic trigonometric Bézier curve are estimated by variational approach to reproduce a circular arc. The radius error is calculated and found less than the existing techniques.

**Keywords:**
Control points,
rational trigonometric Bézier curves,
radius error,
shape measure,
weight functions.

##### 57 Entropy based Expeditive Methodology for Rating Curves Assessment

**Authors:**
D. Mirauda,
M. Greco,
P. Moscarelli

**Abstract:**

The river flow forecasting represents a crucial point to employ for improving a management policy addressed to the right use of water resources as well as for conjugating prevention and defense actions against environmental degradation. The difficulties occurring during the field activities encourage the development and implementation of operative computation and measuring methods addressed to time reduction for data acquisition and processing maintaining a good level of accuracy. Therefore, the aim of the present work is to test a new entropy based expeditive methodology for the evaluation of the rating curves on three gauged sections with different geometric and morphological characteristics. The methodology requires the choice of only three verticals along the measure section and the sampling of only the maximum velocity. The results underline how in most conditions the rating curves drawn can replace those built with classic methodologies, simplifying thus the procedures of data monitoring and calculation.

**Keywords:**
gauged station,
entropic approach,
expeditive
methodology,
rating curves.

##### 56 Positive Definite Quadratic Forms, Elliptic Curves and Cubic Congruences

**Authors:**
Ahmet Tekcan

**Abstract:**

**Keywords:**
Binary quadratic form,
elliptic curves,
cubic congruence.

##### 55 Rational Points on Elliptic Curves 2 3 3y = x + a inF , where p 5(mod 6) is Prime

**Authors:**
Gokhan Soydan,
Musa Demirci,
Nazli Yildiz Ikikardes,
Ismail Naci Cangul

**Abstract:**

In this work, we consider the rational points on elliptic curves over finite fields Fp where p ≡ 5 (mod 6). We obtain results on the number of points on an elliptic curve y2 ≡ x3 + a3(mod p), where p ≡ 5 (mod 6) is prime. We give some results concerning the sum of the abscissae of these points. A similar case where p ≡ 1 (mod 6) is considered in [5]. The main difference between two cases is that when p ≡ 5 (mod 6), all elements of Fp are cubic residues.

**Keywords:**
Elliptic curves over finite fields,
rational points.

##### 54 Improved of Elliptic Curves Cryptography over a Ring

**Authors:**
A. Chillali,
A. Tadmori,
M. Ziane

**Abstract:**

In this article we will study the elliptic curve defined over the ring An and we define the mathematical operations of ECC, which provides a high security and advantage for wireless applications compared to other asymmetric key cryptosystem.

**Keywords:**
Elliptic Curves,
Finite Ring,
Cryptography.

##### 53 A Method to Calculate Frenet Apparatus of W-Curves in the Euclidean 6-Space

**Authors:**
Süha Yılmaz,
Melih Turgut

**Abstract:**

**Keywords:**
Classical Differential Geometry,
Euclidean 6-space,
Frenet Apparatus of the curves.

##### 52 The Number of Rational Points on Elliptic Curves y2 = x3 + a3 on Finite Fields

**Authors:**
Musa Demirci,
Nazlı Yıldız İkikardeş,
Gökhan Soydan,
İsmail Naci Cangül

**Abstract:**

**Keywords:**
Elliptic curves over finite fields,
rational points,
quadratic residue.

##### 51 On Frenet-Serret Invariants of Non-Null Curves in Lorentzian Space L5

**Authors:**
Melih Turgut,
José Luis López-Bonilla,
Süha Yılmaz

**Abstract:**

The aim of this paper is to determine Frenet-Serret invariants of non-null curves in Lorentzian 5-space. First, we define a vector product of four vectors, by this way, we present a method to calculate Frenet-Serret invariants of the non-null curves. Additionally, an algebraic example of presented method is illustrated.

**Keywords:**
Lorentzian 5-space,
Frenet-Serret Invariants,
Nonnull
Curves

##### 50 Contributions to Differential Geometry of Pseudo Null Curves in Semi-Euclidean Space

**Authors:**
Melih Turgut,
Süha Yılmaz

**Abstract:**

**Keywords:**
Semi-Euclidean Space,
Pseudo Null Curves,
Position Vectors.

##### 49 Elliptic Divisibility Sequences over Finite Fields

**Authors:**
Betül Gezer,
Ahmet Tekcan,
Osman Bizim

**Abstract:**

**Keywords:**
Elliptic divisibility sequences,
singular elliptic divisibilitysequences,
elliptic curves,
singular curves.

##### 48 Regionalization of IDF Curves with L-Moments for Storm Events

**Authors:**
Noratiqah Mohd Ariff,
Abdul Aziz Jemain,
Mohd Aftar Abu Bakar

**Abstract:**

The construction of Intensity-Duration-Frequency (IDF) curves is one of the most common and useful tools in order to design hydraulic structures and to provide a mathematical relationship between rainfall characteristics. IDF curves, especially those in Peninsular Malaysia, are often built using moving windows of rainfalls. However, these windows do not represent the actual rainfall events since the duration of rainfalls is usually prefixed. Hence, instead of using moving windows, this study aims to find regionalized distributions for IDF curves of extreme rainfalls based on storm events. Homogeneity test is performed on annual maximum of storm intensities to identify homogeneous regions of storms in Peninsular Malaysia. The L-moment method is then used to regionalized Generalized Extreme Value (GEV) distribution of these annual maximums and subsequently. IDF curves are constructed using the regional distributions. The differences between the IDF curves obtained and IDF curves found using at-site GEV distributions are observed through the computation of the coefficient of variation of root mean square error, mean percentage difference and the coefficient of determination. The small differences implied that the construction of IDF curves could be simplified by finding a general probability distribution of each region. This will also help in constructing IDF curves for sites with no rainfall station.

**Keywords:**
IDF curves,
L-moments,
regionalization,
storm events.

##### 47 Arabic Character Recognition Using Regression Curves with the Expectation Maximization Algorithm

**Authors:**
Abdullah A. AlShaher

**Abstract:**

In this paper, we demonstrate how regression curves can be used to recognize 2D non-rigid handwritten shapes. Each shape is represented by a set of non-overlapping uniformly distributed landmarks. The underlying models utilize 2^{nd} order of polynomials to model shapes within a training set. To estimate the regression models, we need to extract the required coefficients which describe the variations for a set of shape class. Hence, a least square method is used to estimate such modes. We then proceed by training these coefficients using the apparatus Expectation Maximization algorithm. Recognition is carried out by finding the least error landmarks displacement with respect to the model curves. Handwritten isolated Arabic characters are used to evaluate our approach.

**Keywords:**
Shape recognition,
Arabic handwritten characters,
regression curves,
expectation maximization algorithm.

##### 46 A Study of General Attacks on Elliptic Curve Discrete Logarithm Problem over Prime Field and Binary Field

**Authors:**
Tun Myat Aung,
Ni Ni Hla

**Abstract:**

**Keywords:**
Discrete logarithm problem,
general attacks,
elliptic curves,
strong curves,
prime field,
binary field,
attack experiments.

##### 45 A New Method for Computing the Inverse Ideal in a Coordinate Ring

**Authors:**
Abdolali Basiri

**Abstract:**

In this paper we present an efficient method for inverting an ideal in the ideal class group of a Cab curve by extending the method which is presented in [3]. More precisely we introduce a useful generator for the inverse ideal as a K[X]-module.

**Keywords:**
Cab Curves,
Ideal Class Group

##### 44 Development of Machinable Ellipses by NURBS Curves

**Authors:**
Yuan L. Lai,
Jian H. Chen,
Jui P. Hung

**Abstract:**

**Keywords:**
Ellipse,
Approximation,
NURBS,
Optimum.

##### 43 The Control Vector Scheme for Design of Planar Primitive PH curves

**Authors:**
Ching-Shoei Chiang,
Sheng-Hsin Tsai,
James Chen

**Abstract:**

**Keywords:**
PH curve,
hodograph,
Bézier curve.

##### 42 Dynamic Optimization of Industrial Servomechanisms using Motion Laws Based On Bezier Curves

**Authors:**
Giovanni Incerti

**Abstract:**

The motion planning procedure described in this paper has been developed in order to eliminate or reduce the residual vibrations of electromechanical positioning systems, without augmenting the motion time (usually imposed by production requirements), nor introducing overtime for vibration damping. The proposed technique is based on a suitable choice of the motion law assigned to the servomotor that drives the mechanism. The reference profile is defined by a Bezier curve, whose shape can be easily changed by modifying some numerical parameters. By means of an optimization technique these parameters can be modified without altering the continuity conditions imposed on the displacement and on its time derivatives at the initial and final time instants.

**Keywords:**
Servomechanism,
residual vibrations,
motion optimization.

##### 41 Deniable Authentication Protocol Resisting Man-in-the-Middle Attack

**Authors:**
Song Han,
Wanquan Liu,
Elizabeth Chang

**Abstract:**

**Keywords:**
Deniable Authentication,
Man-in-the-middleAttack,
Cryptography,
Elliptic Curves.

##### 40 Simulation of Kinetic Friction in L-Bending of Sheet Metals

**Authors:**
Maziar Ramezani,
Thomas Neitzert,
Timotius Pasang

**Abstract:**

This paper aims at experimental and numerical investigation of springback behavior of sheet metals during L-bending process with emphasis on Stribeck-type friction modeling. The coefficient of friction in Stribeck curve depends on sliding velocity and contact pressure. The springback behavior of mild steel and aluminum alloy 6022-T4 sheets was studied experimentally and using numerical simulations with ABAQUS software with two types of friction model: Coulomb friction and Stribeck friction. The influence of forming speed on springback behavior was studied experimentally and numerically. The results showed that Stribeck-type friction model has better results in predicting springback in sheet metal forming. The FE prediction error for mild steel and 6022-T4 AA is 23.8%, 25.5% respectively, using Coulomb friction model and 11%, 13% respectively, using Stribeck friction model. These results show that Stribeck model is suitable for simulation of sheet metal forming especially at higher forming speed.

**Keywords:**
Friction,
L-bending,
Springback,
Stribeck curves.