Elliptic Curve Cryptography (ECC)
Mathematical basis of ECC
Elliptic Curve is a set of solutions (x, y) to an equation of the form y2=x3+ax+b where 4a3+27b2≠0, together with a point at infinity denoted O. Elliptic Curve originally developed to measure circumference of an ellipse and now have been proposed for applications in cryptography due to their group law and because so far no sub exponential attack on their discrete logarithm problem. Cryptography based on elliptic curves depends on arithmetic involving the points of the curve. Definition: An elliptic curve E over a field K is defined by following equation which is called Weiestress equation. E:y2+a1xy+a3y=x3+a2x2+a4x+a6

y2=x3+ax+b is the simplified version of the Weiestress equation.

Figure 1. Group law on elliptic curve y2=fx over R
Group Law
The definition of Group Law is where the chord-and-tangent rule of adding two points in the curve to give third point which reflects across the x-axis. It is this group that is used in the construction of elliptic curve cryptographic systems. Closure, Inverse, Commutative, Identity and Associativity are conditions that the set and operation must satisfy to be qualify as a group which also known as group axioms. Addition Formulae

Let P1=(x1,y1) and P2=(x2,y2) be non-inverses. Then P1+P2=(x3,y3) Scalar multiplication
Scalar multiplication is repeated group addition: cP=P+…+P (c times)where c is an integer The Elliptic Curve Discrete Logarithm Problem (ECDLP)
The security of all ECC schemes are depends on the hardness of the elliptic curve discrete logarithm problem. Problem: Given two points W, G find s such that W=sG
The elliptic curve parameters for cryptographic schemes should be carefully chosen with appropriate cryptographic restriction in order to resist all known attacks on the ECDLP which is believed to take exponential time. O(sqrtr) time, where r is the order of W

By comparison, factoring and ordinary discrete logarithms can be solved in sub...

...introduction to ellipticcurvecryptography (ECC) and how it is used in the implementation of digital signature (ECDSA) and key agreement (ECDH) Algorithms. The paper discusses the implementation of ECC on two finite fields, prime field and binary field. It also gives an overview of ECC implementation on different coordinate systems called the projective coordinate systems. The paper also discusses the basics...

...EllipticCurves in Public Key Cryptography: The Diffie Hellman Key Exchange Protocol and its relationship to the EllipticCurve Discrete Logarithm Problem Public Key Cryptography Public key cryptography is a modern form of cryptography that allows different parties to exchange information securely over an insecure network, without having first to agree upon some secret key. The main...

...EllipticCurveCryptography and Its Applications to Mobile Devices.
Wendy Chou, University of Maryland, College Park. Advisor: Dr. Lawrence Washington, Department of Mathematics Abstract: The explosive growth in the use of mobile and wireless devices demands a new generation of PKC schemes that has to accommodate limitations on power and bandwidth, at the same time, to provide an adequate level of security for such devices. This paper examines the...

...DATA ENCRYPTION AND DECRYPTION WITH ELLIPTICCURVECRYPTOGRAPHY
Ravi Kumar Kodali
Department of ECE National Institute of Technology Warangal,506004,INDIA
Abstract—Ellipticcurvecryptography is the most popular cryptographic technique that provides higher security with a lesser key size.ECC Encryption and Decryption are implemented through Ellipticcurve points operations like...

...P113
EllipticCurveCryptography
|Vikram V Kumar (vikramv@asu.edu) |[Grad.] |
|Satish Doraiswamy (satish.d@asu.edu) |[Grad.] |
|Zabeer Jainullabudeen (zabeer@asu.edu) |[Grad.] |
Final Report
Abstract
The idea of information security lead to the evolution of...

...settings.
Denmark has a strong historical background in relation to the training of early care and education practitioners. The first training programme was implemented in 1885 for those working from the frobelian approach in education; this formed a basis for the two year study programme implemented in 1904. In 1992, an integrated training system was established to combine theory with practice. Those training to be social pedagogues complete three and a half year degree...

...HR in ECC Division
ECC-ites go through a process of continuous learning, assisted by training programmes. Apart from on-the-job training and technical training, over 100 programmes on general management and behavioural topics are conducted each year. Interactive CD-ROM based programmes have enabled employees learn at their pace.
ECC recognises that people are the real source of competitive advantage. It is through people that...

...Assignment Requirements
Part 1:
Answer the following questions by filling in the blank using the proper technology or cryptography terms you have learned in class:
1. SSL certificates type of certificate is used to provide security on Web sites.
2. PKI is the most common public key encryption systems and, in most cases, this relies on manual trust and key distribution.
3. Message authentication code provides authentication or proves integrity of a digital message....