**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**2476

# Search results for: Elliptic curves over finite fields

##### 2476 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.

##### 2475 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.

##### 2474 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.

##### 2473 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.

##### 2472 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.

##### 2471 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.

##### 2470 On The Elliptic Divisibility Sequences over Finite Fields

**Authors:**
Osman Bizim

**Abstract:**

**Keywords:**
Elliptic divisibility sequences,
equivalent sequences,
singular sequences.

##### 2469 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.

##### 2468 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.

##### 2467 Positive Definite Quadratic Forms, Elliptic Curves and Cubic Congruences

**Authors:**
Ahmet Tekcan

**Abstract:**

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

##### 2466 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.

##### 2465 New DES based on Elliptic Curves

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

**Abstract:**

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

##### 2464 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.

##### 2463 Cryptography Over Elliptic Curve Of The Ring Fq[e], e4 = 0

**Authors:**
Chillali Abdelhakim

**Abstract:**

Groups where the discrete logarithm problem (DLP) is believed to be intractable have proved to be inestimable building blocks for cryptographic applications. They are at the heart of numerous protocols such as key agreements, public-key cryptosystems, digital signatures, identification schemes, publicly verifiable secret sharings, hash functions and bit commitments. The search for new groups with intractable DLP is therefore of great importance.The goal of this article is to study elliptic curves over the ring Fq[], with Fq a finite field of order q and with the relation n = 0, n ≥ 3. The motivation for this work came from the observation that several practical discrete logarithm-based cryptosystems, such as ElGamal, the Elliptic Curve Cryptosystems . In a first time, we describe these curves defined over a ring. Then, we study the algorithmic properties by proposing effective implementations for representing the elements and the group law. In anther article we study their cryptographic properties, an attack of the elliptic discrete logarithm problem, a new cryptosystem over these curves.

**Keywords:**
Elliptic Curve Over Ring,
Discrete Logarithm Problem.

##### 2462 The Number of Rational Points on Singular Curvesy 2 = x(x - a)2 over Finite Fields Fp

**Authors:**
Ahmet Tekcan

**Abstract:**

**Keywords:**
Singular curve,
elliptic curve,
rational points.

##### 2461 An Attack on the Lucas Based El-Gamal Cryptosystem in the Elliptic Curve Group Over Finite Field Using Greater Common Divisor

**Authors:**
Lee Feng Koo,
Tze Jin Wong,
Pang Hung Yiu,
Nik Mohd Asri Nik Long

**Abstract:**

Greater common divisor (GCD) attack is an attack that relies on the polynomial structure of the cryptosystem. This attack required two plaintexts differ from a fixed number and encrypted under same modulus. This paper reports a security reaction of Lucas Based El-Gamal Cryptosystem in the Elliptic Curve group over finite field under GCD attack. Lucas Based El-Gamal Cryptosystem in the Elliptic Curve group over finite field was exposed mathematically to the GCD attack using GCD and Dickson polynomial. The result shows that the cryptanalyst is able to get the plaintext without decryption by using GCD attack. Thus, the study concluded that it is highly perilous when two plaintexts have a slight difference from a fixed number in the same Elliptic curve group over finite field.

**Keywords:**
Decryption,
encryption,
elliptic curve,
greater common divisor.

##### 2460 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.

##### 2459 The Number of Rational Points on Conics Cp,k : x2 − ky2 = 1 over Finite Fields Fp

**Authors:**
Ahmet Tekcan

**Abstract:**

Let p be a prime number, Fp be a finite field, and let k ∈ F*p. In this paper, we consider the number of rational points onconics Cp,k: x2 − ky2 = 1 over Fp. We proved that the order of Cp,k over Fp is p-1 if k is a quadratic residue mod p and is p + 1 if k is not a quadratic residue mod p. Later we derive some resultsconcerning the sums ΣC[x]p,k(Fp) and ΣC[y]p,k(Fp), the sum of x- and y-coordinates of all points (x, y) on Cp,k, respectively.

**Keywords:**
Elliptic curve,
conic,
rational points.

##### 2458 The Diophantine Equation y2 − 2yx − 3 = 0 and Corresponding Curves over Fp

**Authors:**
Ahmet Tekcan,
Arzu Özkoç,
Hatice Alkan

**Abstract:**

**Keywords:**
Diophantine equation,
Pell equation,
quadratic form.

##### 2457 Nonlinear Static Analysis of Laminated Composite Hollow Beams with Super-Elliptic Cross-Sections

**Authors:**
G. Akgun,
I. Algul,
H. Kurtaran

**Abstract:**

In this paper geometrically nonlinear static behavior of laminated composite hollow super-elliptic beams is investigated using generalized differential quadrature method. Super-elliptic beam can have both oval and elliptic cross-sections by adjusting parameters in super-ellipse formulation (also known as Lamé curves). Equilibrium equations of super-elliptic beam are obtained using the virtual work principle. Geometric nonlinearity is taken into account using von-Kármán nonlinear strain-displacement relations. Spatial derivatives in strains are expressed with the generalized differential quadrature method. Transverse shear effect is considered through the first-order shear deformation theory. Static equilibrium equations are solved using Newton-Raphson method. Several composite super-elliptic beam problems are solved with the proposed method. Effects of layer orientations of composite material, boundary conditions, ovality and ellipticity on bending behavior are investigated.

**Keywords:**
Generalized differential quadrature,
geometric nonlinearity,
laminated composite,
super-elliptic cross-section.

##### 2456 Scalable Systolic Multiplier over Binary Extension Fields Based on Two-Level Karatsuba Decomposition

**Authors:**
Chiou-Yng Lee,
Wen-Yo Lee,
Chieh-Tsai Wu,
Cheng-Chen Yang

**Abstract:**

Shifted polynomial basis (SPB) is a variation of polynomial basis representation. SPB has potential for efficient bit level and digi -level implementations of multiplication over binary extension fields with subquadratic space complexity. For efficient implementation of pairing computation with large finite fields, this paper presents a new SPB multiplication algorithm based on Karatsuba schemes, and used that to derive a novel scalable multiplier architecture. Analytical results show that the proposed multiplier provides a trade-off between space and time complexities. Our proposed multiplier is modular, regular, and suitable for very large scale integration (VLSI) implementations. It involves less area complexity compared to the multipliers based on traditional decomposition methods. It is therefore, more suitable for efficient hardware implementation of pairing based cryptography and elliptic curve cryptography (ECC) in constraint driven applications.

**Keywords:**
Digit-serial systolic multiplier,
elliptic curve
cryptography (ECC),
Karatsuba algorithm (KA),
shifted polynomial
basis (SPB),
pairing computation.

##### 2455 A Pairing-based Blind Signature Scheme with Message Recovery

**Authors:**
Song Han,
Elizabeth Chang

**Abstract:**

Blind signatures enable users to obtain valid signatures for a message without revealing its content to the signer. This paper presents a new blind signature scheme, i.e. identity-based blind signature scheme with message recovery. Due to the message recovery property, the new scheme requires less bandwidth than the identitybased blind signatures with similar constructions. The scheme is based on modified Weil/Tate pairings over elliptic curves, and thus requires smaller key sizes for the same level of security compared to previous approaches not utilizing bilinear pairings. Security and efficiency analysis for the scheme is provided in this paper.

**Keywords:**
Blind Signature,
Message Recovery,
Pairings,
Elliptic Curves,
Blindness

##### 2454 Numerical Study of Liquefied Petroleum Gas Laminar Flow in Cylindrical Elliptic Pipes

**Authors:**
Olumuyiwa A. Lasode,
Tajudeen O. Popoola,
B. V. S. S. S. Prasad

**Abstract:**

Fluid flow in cylinders of elliptic cross-section was investigated. Fluid used is Liquefied petroleum gas (LPG). LPG found in Nigeria contains majorly butane with percentages of propane. Commercial available code FLUENT which uses finite volume method was used to solve fluid flow governing equations. There has been little attention paid to fluid flow in cylindrical elliptic pipes. The present work aims to predict the LPG gas flow in cylindrical pipes of elliptic cross-section. Results of flow parameters of velocity and pressure distributions are presented. Results show that the pressure drop in elliptic pipes is higher than circular pipe of the same cross-sectional area. This is an important result as the pressure drop is related to the pump power needed to drive the flow. Results show that the velocity increases towards centre of the pipe as the flow moves downstream, and also increases towards the outlet of the pipe.

**Keywords:**
Elliptic Pipes,
Liquefied Petroleum Gas,
Numerical Study,
Pressure Drop.

##### 2453 Experimental and Numerical Study of The Shock-Accelerated Elliptic Heavy Gas Cylinders

**Authors:**
Jing S. Bai,
Li Y. Zou,
Tao Wang,
Kun Liu,
Wen B. Huang,
Jin H. Liu,
Ping Li,
Duo W. Tan,
CangL. Liu

**Abstract:**

**Keywords:**
About four key words or phrases in alphabeticalorder,
separated by commas.

##### 2452 Efficient Hardware Implementation of an Elliptic Curve Cryptographic Processor Over GF (2 163)

**Authors:**
Massoud Masoumi,
Hosseyn Mahdizadeh

**Abstract:**

A new and highly efficient architecture for elliptic curve scalar point multiplication which is optimized for a binary field recommended by NIST and is well-suited for elliptic curve cryptographic (ECC) applications is presented. To achieve the maximum architectural and timing improvements we have reorganized and reordered the critical path of the Lopez-Dahab scalar point multiplication architecture such that logic structures are implemented in parallel and operations in the critical path are diverted to noncritical paths. With G=41, the proposed design is capable of performing a field multiplication over the extension field with degree 163 in 11.92 s with the maximum achievable frequency of 251 MHz on Xilinx Virtex-4 (XC4VLX200) while 22% of the chip area is occupied, where G is the digit size of the underlying digit-serial finite field multiplier.

**Keywords:**
Elliptic curve cryptography,
FPGA implementation,
scalar point multiplication.

##### 2451 Maximum Norm Analysis of a Nonmatching Grids Method for Nonlinear Elliptic Boundary Value Problem −Δu = f(u)

**Authors:**
Abida Harbi

**Abstract:**

**Keywords:**
Error estimates,
Finite elements,
Nonlinear PDEs,
Schwarz method.

##### 2450 MEGSOR Iterative Scheme for the Solution of 2D Elliptic PDE's

**Authors:**
J. Sulaiman,
M. Othman,
M. K. Hasan

**Abstract:**

Recently, the findings on the MEG iterative scheme has demonstrated to accelerate the convergence rate in solving any system of linear equations generated by using approximation equations of boundary value problems. Based on the same scheme, the aim of this paper is to investigate the capability of a family of four-point block iterative methods with a weighted parameter, ω such as the 4 Point-EGSOR, 4 Point-EDGSOR, and 4 Point-MEGSOR in solving two-dimensional elliptic partial differential equations by using the second-order finite difference approximation. In fact, the formulation and implementation of three four-point block iterative methods are also presented. Finally, the experimental results show that the Four Point MEGSOR iterative scheme is superior as compared with the existing four point block schemes.

**Keywords:**
MEG iteration,
second-order finite difference,
weighted parameter.

##### 2449 Proposed Developments of Elliptic Curve Digital Signature Algorithm

**Authors:**
Sattar B. Sadkhan,
Najlae Falah Hameed

**Abstract:**

**Keywords:**
Elliptic Curve Digital Signature Algorithm,
DSA.

##### 2448 Solving Stochastic Eigenvalue Problem of Wick Type

**Authors:**
Hassan Manouzi,
Taous-Meriem Laleg-Kirati

**Abstract:**

In this paper we study mathematically the eigenvalue problem for stochastic elliptic partial differential equation of Wick type. Using the Wick-product and the Wiener-Itô chaos expansion, the stochastic eigenvalue problem is reformulated as a system of an eigenvalue problem for a deterministic partial differential equation and elliptic partial differential equations by using the Fredholm alternative. To reduce the computational complexity of this system, we shall use a decomposition method using the Wiener-Itô chaos expansion. Once the approximation of the solution is performed using the finite element method for example, the statistics of the numerical solution can be easily evaluated.

**Keywords:**
Eigenvalue problem,
Wick product,
SPDEs,
finite
element,
Wiener-Itô chaos expansion.

##### 2447 Magnetic Field Analysis for a Distribution Transformer with Unbalanced Load Conditions by using 3-D Finite Element Method

**Authors:**
P. Meesuk,
T. Kulworawanichpong,
P. Pao-la-or

**Abstract:**

**Keywords:**
Distribution Transformer,
Magnetic Field,
Load
Unbalance,
3-D Finite Element Method (3-D FEM)