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How to cite this article: Kefa Rabah , Theory and Implementation of Elliptic Curve Cryptography. Journal of Applied Sciences, 5: However the need to exchange information securely is still very important and is therefore provided in electronic documents; usually by encryption and digital signature s. The science of keeping messages secure is called cryptography. Cryptography involves encryption and decryption of messages. Encryption is the process of converting a plaintext into ciphertext by using an algorithm and decryption is the process of getting back the encrypted message Fig.

A cryptographic algorithm is the mathematical function used for encryption and decryption. In addition to providing confidentiality, cryptography is often required to provide Authentication, Integrity and Non-repudiation.

The essence of cryptography is traditionally captured in the following problem: Two parties the tradition is to call them Bob and Alice wish to communicate over an insecure public communication channel in the presence of malevolent eavesdropper the tradition Eve.

Bob and Alice could be military jets, e-business or just friend trying to have a private conversation Fig. Moreover, it is worthy to note right from the start that the hardest part of computer security is the piece between the computer and the user. In case of the digital signature s, the hardest part is proving that the text signed is the same text that the user viewed.

And finally the hardest part of computer forensics is to know who is sitting in front of a particular computer at any time. There are two popular kinds of cryptographic protocols: symmetric-key and asymmetric-key protocols. In the symmetric-key protocols, a common key the secret-key is used by both communicating partners to encrypt and decrypt messages[1].

These symmetric-key cryptosystems provide high speed key and communication but have the drawback that a common or session key must be established for each pair of participants[2]. The encoding function here is a trapdoor function-one whose inverse is impractical to implement, unless some extra information is available. This extra information called the decrypting-key is not required for encrypting the message, yet is essential for decrypting it in reasonable time.

This makes it much easier to encrypt messages than to decrypt them. The beauty of such a system is that the encrypting process need not be kept secret. Each user has his own or a personal encrypting-function, which is public information hence the name Public-Key and a decoding key, which he keeps secret. One of these keys the public-key is used either for encryption signature of the messages and; a different key the private-key is used for either decryption confidentiality of the message[4].

Different public-key cryptographic systems are used to provide public-key security. These systems provide these services by relying on the difficulty of different classical mathematical problems, hence provide the services in different ways.

The public-key cryptosystems are, however, slower than the symmetric ones but provide arbitrary high levels of security and do not require an initial private key exchange between two communicating parties.

In the asymmetric protocol the public-key is released to the public while the other, the private-key, is known only to its owner. Because of this feature, these cryptosystems are considered to be indispensable for secure communication and authentication over open insecure networks[9]. It is designed to be computationally intractable to calculate a private-key from its associated public-key; that is, it is believed that any attempt to compute it will fail even when up-to-date technology and equipment are used[10].

Therefore, they are suitable for communication among the general public. Public-key cryptosystems can also be used to make a digital signature [11]. However, in real applications, both symmetric and asymmetric protocols are used. The public-key algorithm is first used for establishing a common symmetric-key over insecure channel. Then the symmetric system is used for secure communication with high throughput.

Due to comparative slowness of the public-key algorithms, dedicated hardware is desirable for efficient implementation and operation of the cryptographic systems. In the mid s researchers noticed that another source of hard problems might be discovered by looking at the elliptic curves[12,13]. The invention of Elliptic Curve Cryptography ECC offered a new level of security for public key cryptosystems[], which provide both encryption and digital signature s services.

Furthermore, elliptic curve can provide versions of public-key methods that, in some cases, are faster and use smaller keys, while providing an equivalent level of security.

Their advantage comes from using different kind of mathematical group for public-key arithmetic. To date many research papers in Elliptic Curve Cryptography ECC have been published by researchers all over the world, as can be viewed in the refs. However, the idea of using elliptic curves in cryptography is still considered a difficult concept and is neither widely accepted nor understood by typical technical people.

The problem may stem from the fact that there is a large gap between the theoretical mathematics of elliptic curves and the applications of elliptic curves in cryptography. Further, it is important to note that, however, it is easy to calculate the surface of the ellipse, it is hard to calculate the circumference of the ellipse. This was done already during the 18th century. Probably, it was Abel in ca. At the moment there are several definitions for an elliptic curve.

However, the following definition is used usually: Definition: An elliptic curves E, defined over an arbitrary field K, is a non-supersingular plain projective third degree curve over K with a K-rational point O i. This discriminant is a polynomial expression in the coefficient a1 , This point plays the role of O origin. Sometimes we want to express that a curve E is based over field K. Remark: For practical applications finite field of the form are very important.

For such elliptic curves the theory as mentioned above has to be modified. In , Yutaka Taniyama asked some questions about elliptic curves, i. Elliptic curves can also be looked at as mathematical constructions from number theory and algebraic geometry, which in recent years have found numerous applications in cryptography[12].

Elliptic curve cryptosystem ECC is relatively new. The ECC was first introduced by Miller[13] and independently by Koblitz[12] in the mid s and today it has evolved into a mature public-key cryptosystem. It was also recently endorsed by the U. This is so, because elliptic curves do not introduce new cryptographic algorithms, but they implement existing public-key algorithms using elliptic curves.

In this way, variants of existing schemes can be devised that rely for their security on a different underlying hard problem. Elliptic curve scheme: An elliptic curve can be defined over any field e. However, elliptic curves used in cryptography are mainly defined over finite fields[21]. Addition of two points on an elliptic curve is defined according to a set of simple rules e. The addition operation in an elliptic curve is the counterpart to modular multiplication in common public-key cryptosystems and multiple addition is the counterpart to modular exponentiation[22,23], as will be seen later.

Here, the underlying field of integers modulo prime p is replaced by points on an elliptic curve defined over a finite field. Elliptic curves have further shown themselves to be remarkably useful in a range of applications including primality testing and integer factorization[]. Elliptic Curve Cryptosystems ECCs also include key distribution, encryption and digital signature algorithms.

The key distribution algorithm is used to share a secret-key, the encryption algorithm enables confidential communication and the digital signature algorithm is used to authenticate the signer and validate the integrity of the message[28,29].

The mechanics of finite field Fp: Abstractly a finite field consists of a finite set of objects called field elements F together with the description of two operations-addition and multiplication-that can be performed on pairs of field elements. These operations must possess certain properties.

It turns out that there is a finite field containing q field elements if and only if q is a power of a prime number and furthermore that in fact for each such q there is precisely one finite field. The finite field containing q elements is denoted by Fq. The order of a finite field is the number of elements in the field. There exists a finite field of order q if and only if q is a prime power, then there are, however, many efficient implementations of the field arithmetic in hardware or in software[12,15,21,30,31].

If in adding the multiplicative identity 1 to itself in F never gives 0, then we say that F has characteristic zero; in that case F contains a copy of the field of rotational numbers. This study describe the elements, operations and implementation of the finite field Fp, while the elements and operations of can be found elsewhere[33,34].

Let p be a prime number. This is known as addition modulo p. This is known as multiplication modulo p. The mechanics of elliptic curves: An elliptic curve over Fq is defined in terms of the solutions to an equation in Fq. The form of the equation defining an elliptic curve over Fq differs depending on whether the field is a prime finite field or a characteristic 2 finite field. Roughly speaking, an elliptic curve is the solution set of a cubic equation in two variables.

The solution set is relative to some field of definition, such as the field of complex numbers. Of greatest interest are the rational solutions of such equations.

The rational points on the elliptic curve E are the points over Ep a, b that satisfy the defining equation. If the set of parameters a, b, p are specified, the number of rational points on the elliptic curve is determined uniquely; this number is called the order of the elliptic curve E and is denoted by E. It is known that rational points form an additive group in the addition over the elliptic curve shown in Fig.

But much of the theory-and essentially all the cryptographic applications-lie in the solutions mod p or, more generally, in solutions over finite fields. One of the nice features of elliptic curves mod p is that the size of the solution set is never too far from p.

This ensures that for large p, there are lots of solutions over the finite field Fp. In many cases, the order of the group over Fp.

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## Theory and Implementation of Elliptic Curve Cryptography

Daigor Kefa Rabah Competitiveness is an interesting subject nowadays, and today has permeated from local to global business of all sorts. Rabah Kefa Kenya, similar to other Africa nations, is on the move, and ICT offers powerful krfa to accelerate this therein boosting economic growth and poverty reduction. Short Message Service as Key for Steganography. From organizational resources undertaking: Reviews Schrijf een review. Levertijd We doen er alles aan om dit artikel op tijd te bezorgen. Samenvatting Competitiveness is an interesting subject nowadays, and today has permeated from local to global business of all sorts. It support different protocols for providing communication using different tools, e.

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