The Binary Operations Calculus in

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.

IACR Cryptol. ePrint Arch., 2010

This paper presents a new model of ordinary elliptic curves with fast arithmetic over field of characteristic two. In addition, we propose two isomorphism maps between new curves and Weierstrass curves. This paper proposes new explicit addition law for new binary curves and prove the addition law corresponds to the usual addition law on Weierstrass curves. This paper also presents fast unified addition formulae and doubling formulae for these curves. The unified addition formulae cost 12M +2D, where M is the cost of a field multiplication, and D is the cost of multiplying by a curve parameter. These formulae are more efficient than other formulae in literature. Finally, this paper presents explicit formulae for w-coordinates differential addition. In a basic step of Montgomery ladder, the cost of a projective differential addition and doubling are 5M and 1M +1D respectively, and the cost of mixed w-coordinates differential addition is 4M .

Zenodo (CERN European Organization for Nuclear Research), 2011

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.

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.

, M.Virat has established the group law on the elliptic curve defined over the ring F p d [ε], ε n = 0, where p a prime number 5, d ∈ N * and n = 2; and in [4] A.Chillali has, on his part studied the same case for p and extend it to n = 3.

2008

En els darrers anys, la criptografia amb corbes el.líptiques ha adquirit una importància creixent, fins a arribar a formar part en la actualitat de diferents estàndards industrials. Tot i que s'han dissenyat variants amb corbes el.líptiques de criptosistemes clàssics, com el RSA, el seu màxim interès rau en la seva aplicació en criptosistemes basats en el Problema del Logaritme Discret,

Undergraduate Texts in Mathematics, 2014

Elliptic curves have been intensively studied in number theory and algebraic geometry for over 100 years and there is an enormous amount of literature on the subject. To quote the mathematician Serge Lang: It is possible to write endlessly on elliptic curves. (This is not a threat.) Elliptic curves also figured prominently in the recent proof of Fermat's Last Theorem by Andrew Wiles. Originally pursued for purely aesthetic reasons, elliptic curves have recently been utilized in devising algorithms for factoring integers, primality proving, and in public-key cryptography. In this article, we aim to give the reader an introduction to elliptic curve cryptosystems, and to demonstrate why these systems provide relatively small block sizes, high-speed software and hardware implementations, and offer the highest strength-per-key-bit of any known public-key scheme.

In this paper we propose an application of public key distribution based on the security depending on the difficulty of elliptic curve discrete logarithm problem. More precisely, we propose an example of Elgamal encryption cryptosystem on the elliptic curve given by the equation: y 2 = x 3 + 70x + 57[73].

GIS SCIENCE

Algebraic curves over binary and finite fields used in the design of public key cryptography. This paper discusses some topics in algebraic curve cryptography namely elliptic curve cryptography with recent developments and different algorithms in Elliptic curve cryptography and also discussed discrete logarithmic problem and security protocol.

2013

Abstract. The basic operation in elliptic curve cryptosystem is scalar multiplication. It is the computation of integer multiple of a given point on the curve. Computation of scalar multiple is faster by using signed binary representation as compared to binary representation. In this paper ‘Direct Recoding Method ’ a new modified algorithm for computation of signed binary representation is proposed. Our proposed method is efficient when compared to other standard methods such as NAF, MOF and complementary recoding method.