Elliptic Curves and Their Applications to Cryptography

Using Elliptic curves as asubstitute to RSA for signcryption , technique which accomplishes digital signature and encryption in single procedure.

2013

Elliptic curve cryptography is an asymmetric key cryptography. The points on two dimensional elliptic curve are used for declaration of data encryption & decryption. It include public key generation on the elliptic curve and private key generation to decrypt the data. The present paper deals with an overview of Elliptic curve cryptography (ECC) and its implementation through coordinate geometry for data encryption. We introduce a new approach in the form of cardan's method to find points on X axis at elliptic curve over finite field and form public key cryptographic system and finally we define two dimensional alphabetic table and description in the form of algorithm to use it for plain text encryption.

Amazing Fact: We can use geometry to make the points of an elliptic curve into a group. The next few slides illustrate how this is accomplished.

Elliptic curves were introduced to the ancient science of cryptography in the mid 1980s, and Elliptic Curve Cryptography (ECC) has since been growing rapidly. However, owing to the incompleteness of the Weierstrass addition law, elliptic curve cryptosystems based on the Weierstrass model are vulnerable to side-channel attacks. New addition algorithms and elliptic curve models have been proposed to take elliptic curve cryptosystems resistant to side-channel attacks. A promising model in this regard is the Edwards model introduced in 2007. The Edwards addition law is both complete and has the fastest known implementations for elliptic curve operations like addition and doubling. As a part of this work we study the Edwards model in relation to ECC with an emphasis on its computational aspects. We also study two encoding schemes, Elligator and Elligator Square, for representing elliptic curve points as bit strings indistinguishable from uniform random bit strings, both of which have formulations over Edwards curves. We also study isogenies and their computation using analogues of V ́elu’s and Kohel’s formulas for the Edwards model, which turn out to be simpler and more efficient than those for the Weierstrass model. We implement an hitherto unavailable library for Edwards curves, and two ECC algorithms using the implemented Edwards curves, in the mathematical software Sage.

Cryptography is the technique of transforming an intelligible message into unintelligible format so that the message can't be read or understood by an unauthorized person during its transmission over the public networks. A number of cryptographic techniques have been developed over the centuries. With technological advancement, new techniques have been evolved significantly. Public key cryptography offers a great security for transmitting data over the public networks such as Internet. The popular public key cryptosystems like RSA and Diffie-Hellman are becoming slowly disappearing because of requirement of large number of bits in the encryption and decryption keys. Elliptic Curve Cryptograph (ECC) is emerging as an alternative to the existing public key cryptosystems. This paper describes the idea of Elliptic Curve Cryptography (ECC) and its implementation through two dimensional (2D) geometry for data encryption and decryption. This paper discusses the implementation of ECC over prime field. Much attention has been given on the mathematics of elliptic curves starting from their derivations.

Proceedings of the …, 2006

In recent years, elliptic curve cryptography (ECC) has gained widespread exposure and acceptance, and has already been included in many security standards. Engineering of ECC is a complex, interdisciplinary research field encompassing such fields as mathematics, computer science, and electrical engineering. In this paper, we survey ECC implementation issues as a prominent case study for the relatively new discipline of cryptographic engineering. In particular, we show that the requirements of efficiency and security considered at the implementation stage affect not only mere low-level, technological aspects but also, significantly, higher level choices, ranging from finite field arithmetic up to curve mathematics and protocols.

2006

Public key cryptography systems are based on sound mathematical foundations that are designed to make the problem hard for an intruder to break into the system. The major approaches that since 1976 have withstood intruder attacks, are the discrete logarithm Mr. Raja Ghosal PhD Student, Auto-ID Lab, ADELAIDE School of Electrical and Electronics Engineering, The University of Adelaide Prof. Peter H. Cole Research Director, Auto-ID Lab, ADELAIDE School of Electrical and Electronics Engineering, The University of Adelaide Contact: [email protected] or [email protected]. Internet: www.autoidlabs.org

Cambridge University Press eBooks, 1997

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Applied Mathematics [Working Title]

The scalar multiplication on elliptic curves defined over finite fields is a core operation in elliptic curve cryptography (ECC). Several different methods are used for computing this operation. One of them, the binary method, is applied depending on the binary representation of the scalar v in a scalar multiplication vP, where P is a point that lies on elliptic curve E defined over a prime field F p. On the binary method, two methodologies are performed based on the implementation of the binary string bits from the right to the left (RLB) [or from the left to the right (LRB)]. Another method is a nonadjacent form (NAF) which depended on the signed digit representation of a positive integer v. In this chapter, the graphs and subgraphs are employed for the serial computations of elliptic scalar multiplications defined over prime fields. This work proposed using the subgraphs H of the graphs G or the (simple, undirected, directed, connected, bipartite, and other) graphs to represent a scalar v directly. This usage speeds up the computations on the elliptic scalar multiplication algorithms. The computational complexities of the proposed algorithms and previous ones are determined. The comparison results of the computational complexities on all these algorithms are discussed. The experimental results show that the proposed algorithms which are used the sub-graphs H and graphs G need to the less costs for computing vP in compare to previous algorithms which are employed the binary representations or NAF expansion. Thus, the proposed algorithms that use the subgraphs or the graphs to represent the scalars v are more efficient than the original ones.

2015

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

International Journal of Electrical and Computer Engineering (IJECE), 2011

The paper presents an extensive and careful study of elliptic curve cryptography (ECC) and its applications. This paper also discuss the arithmetic involved in elliptic curve and how these curve operations is crucial in determining the performance of cryptographic systems. It also presents different forms of elliptic curve in various coordinate system , specifying which is most widely used and why. It also explains how isogenenies between elliptic curve provides the secure ECC. Exentended form of elliptic curve i.e hyperelliptic curve has been presented here with its pros and cons. Performance of ECC and HEC is also discussed based on scalar multiplication and DLP.

Since the earliest times, individuals and groups of individuals have been interested in communicating sensitive information in a manner which would guarantee that such information could not be arbitrarily received. Further, such information was to be received by select recipients and this required that a means of secure information transmission be found and employed. To these ends, methods of information encryption have ever since been sought and employed. The entire study and practice of this activity, cryptology, the science of message encryption and decryption, provides a framework for this thesis. In particular, the development of cryptology has been influenced by some specific areas of mathematics, employing abstract mathematical concepts and utilizing algebraic structures known as elliptic curves. It is with respect to these structures and their utilization in specific cryptosystems, called elliptic curve cryptosystems on which this thesis focuses. More specifically, this thesis is concerned with the implementation of such a cryptosystem and is a demonstration of that implementation. Additional pertinent examples, illustrations and supporting computer programs are included to present a self-contained work.

Discrete Mathematics and Its Applications, 2005

Background on p-adic Numbers David Lubicz Contents in Brief 3.1 Definition of Qp Qp Qp Qp Qp Qp and first properties 39 3.2 Complete discrete valuation rings and fields 41 First properties • Lifting a solution of a polynomial equation 3.3 The field Qp Qp Qp Qp Qp Qp and its extensions 43 Unramified extensions • Totally ramified extensions • Multiplicative system of representatives • Witt vectors

2011