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...

...Abstract This paper gives an 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 of prime and binary field arithmetic. This paper also discusses why ECC is a better option than RSA in modern day systems. This paper also discusses why ECC's unique properties make it especially well suited to smart card applications.
Index TermsEllipticCurveCryptography, Smart Cards, Discrete Logarithm problem
I. INTRODUCTION
Over the past 30 years, public key cryptography has become a mainstay for secure communications over the Internet and throughout many other forms of communications. It provides the foundation for both key management and digital signatures. In key management, public key cryptography is used to distribute the secret keys used in other cryptographic algorithms (e.g. DES). For digital signatures, public key cryptography is used to authenticate the origin of data and protect the integrity of that data. For the past 20 years,...

...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 use of public key cryptography is to provide information security in computer science, for example to transfer securely email, credit card details or other secret information between sender and recipient via the internet. There are three steps involved in transferring information securely from person A to person B over an insecure network. These are encryption of the original information, called the plaintext, transfer of the encrypted message, or ciphertext, and decryption of the ciphertext back into plaintext. Since the transfer of the ciphertext is over an insecure network, any spy has access to the ciphertext and thus potentially has access to the original information, provided he is able to decipher the message. Thus, a successful cryptosystem must be able encrypt the original message in such a way that only the intended receiver can decipher the ciphertext. The goal of public key cryptography is to make the problem of deciphering the encrypted message...

...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 use of ECC in such constrained environments and discusses the basis of its security, explores its performance and lastly, surveys the use of ECC applications on the market today.
1
Introduction
In 1976, Whitfield Diffie and Martin Hellman introduced the concept of public key cryptography (PKC). Since then, many implementations of it have been proposed, and many of these cryptographic applications base their security on the intractability of hard mathematical problems, namely the integer factorization problem (IFP) and the finite field discrete logarithm problem (DLP). Over the years, sub-exponential time algorithms were developed to solve these problems. As a result, key sizes grew to more than 1000 bits, so as to attain a reasonable level of security. In constrained environments where computing power, storage and bandwidth are limited, carrying out thousand-bit operations becomes an impractical approach to...

...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 point addition,subtraction,doubling and scalar point multiplication.ECC cryptography technique is quite useful for WSN Applications.The security level increases with a larger keysize.The paper focuses on the implementation of ECC encryption and decryption with a key size of 163 bits.
proposed independently by Neal Koblitz and Victor Miller in 1985. They found the ellipticcurve equation while for the arc length of an ellipse. EllipticCurve is deﬁned by the following Weierstrass Equation: y 2 + (a1 xy) + a3 y = x3 + a2 x2 + a4 x + a6 (1)
Keywords: ECC,Encryption,Decryption,Key size I. INTRODUCTION Cryptography is the science and study of secret writing. The Cryptography means scrambling or modifying the information in such a way that it is not understandable and hence information is hidden from an unauthorized person. The main objectives of...

...Analysis of Algorithms
Project ID: 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 Cryptography. In other words, Cryptography is the science of keeping information secure. It involves encryption and decryption of messages. Encryption is the process of converting a plain text into cipher text and decryption is the process of getting back the original message from the encrypted text. Cryptography, in addition to providing confidentiality, also provides Authentication, Integrity and Non-repudiation. The crux of cryptography lies in the key involved and the secrecy of the keys used to encrypt or decrypt. Another important factor is the key strength, i.e. the size of the key so that it is difficult to perform a brute force on the plain and cipher text and retrieve the key. There have been various cryptographic algorithms suggested. In this project we study and analyze the EllipticCurve cryptosystems. This system has been proven to be stronger than known...

...offering cash benefits rather than services especially for the youngest group of children 0-2 years (OECD, 2010).
Pedagogy
French, 2007 defines pedagogy as the practice or craft of teaching in the interactive process between teachers and learners and the learning environment which includes family and the community. Under Siolta, the national quality framework it is defined as the range of interactions to support the holistic development of children by embracing both care and education within 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 programme learning theoretical, culture based subjects and activity based subjects with stints in placement (OECD,2000).
Ireland in comparison has yet to regulate for proper qualifications within the sector with only those working with pre-primary classes having the appropriate qualification – bachelor of education. In order for the pre-school year to be implemented room leaders must have a minimum qualification of a Level 5 major award in ECCE (Dept. of Education &Skills, 2011).
Curriculum...

...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 thatECC delivers total customer satisfaction. These values are reflected in our Human Resources practices which have earned national recognition several times.
ECC has always believed in experimentation with and implementation of new ideas. HR practices such as collaborative performance appraisal, career & succession planning, team rewards have been institutionalised.
An extensive and rigorous recruitment process ensures quality induction. L&T's Graduate Engineer Trainee recruitment process covers India's major engineering colleges and institutions. Programmes, plant visits and comprehensive information-sharing facilitate induction.
ECC Division has an ongoing organization development programme, which is one of the longest sustaining OD effort in India.
Awards for our HR Practices
• Asia Pacific HR Conclave Award for the company with “Most Innovative HR practices”
• National award for “Best HRD Practices” from Indian Society for Training...

...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.
4. SHA-1 encryption scheme was broken and was replaced with a third round version of itself.
5. RSA is the first algorithm suited to both signing and encryption, and it is now widely used in e-commerce and other public key systems.
6. The entity that issues certificates is a Certificate Authority
7. The document to check to verify whether a certificate has been revoked is Certificate Revocation List.
8. Each bit of length determines the number of keys.
9. Currently, 128 bit certificates are commonly used for web communications.
10. Triple DES provides 112 bits of security, despite using a 168 bit key.
11. Thawte, Verisign, and Comodo are all examples of SSL certificates.
12. Hiding data in images is an example of Steganography
13. Data Encryption Standard (DES), ROT13, and Enigma are all examples of algorithm for the encryption of electronic data
14. A digitally signed email provides authentication.
15. A digitally encrypted e-mail provides...