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# Edwards curve

About: Edwards curve is a research topic. Over the lifetime, 815 publications have been published within this topic receiving 25447 citations.

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IBM

^{1}TL;DR: In this paper, an analogue of the Diffie-Hellmann key exchange protocol was proposed, which appears to be immune from attacks of the style of Western, Miller, and Adleman.

Abstract: We discuss the use of elliptic curves in cryptography. In particular, we propose an analogue of the Diffie-Hellmann key exchange protocol which appears to be immune from attacks of the style of Western, Miller, and Adleman. With the current bounds for infeasible attack, it appears to be about 20% faster than the Diffie-Hellmann scheme over GF(p). As computational power grows, this disparity should get rapidly bigger.

4,004 citations

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Hewlett-Packard

^{1}TL;DR: In the past few years elliptic curve cryptography has moved from a fringe activity to a major challenger to the dominant RSA/DSA systems as mentioned in this paper, and it has become all pervasive.

Abstract: In the past few years elliptic curve cryptography has moved from a fringe activity to a major challenger to the dominant RSA/DSA systems. Elliptic curves offer major advances on older systems such as increased speed, less memory and smaller key sizes. As digital signatures become more and more important in the commercial world the use of elliptic curve-based signatures will become all pervasive. This book summarizes knowledge built up within Hewlett-Packard over a number of years, and explains the mathematics behind practical implementations of elliptic curve systems. Due to the advanced nature of the mathematics there is a high barrier to entry for individuals and companies to this technology. Hence this book will be invaluable not only to mathematicians wanting to see how pure mathematics can be applied but also to engineers and computer scientists wishing (or needing) to actually implement such systems.

1,697 citations

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19 Jul 2005

TL;DR: The introduction to Public-Key Cryptography explains the development of algorithms for computing Discrete Logarithms and their applications in Pairing-Based Cryptography and its applications in Fast Arithmetic Hardware Smart Cards.

Abstract: Preface Introduction to Public-Key Cryptography Mathematical Background Algebraic Background Background on p-adic Numbers Background on Curves and Jacobians Varieties Over Special Fields Background on Pairings Background on Weil Descent Cohomological Background on Point Counting Elementary Arithmetic Exponentiation Integer Arithmetic Finite Field Arithmetic Arithmetic of p-adic Numbers Arithmetic of Curves Arithmetic of Elliptic Curves Arithmetic of Hyperelliptic Curves Arithmetic of Special Curves Implementation of Pairings Point Counting Point Counting on Elliptic and Hyperelliptic Curves Complex Multiplication Computation of Discrete Logarithms Generic Algorithms for Computing Discrete Logarithms Index Calculus Index Calculus for Hyperelliptic Curves Transfer of Discrete Logarithms Applications Algebraic Realizations of DL Systems Pairing-Based Cryptography Compositeness and Primality Testing-Factoring Realizations of DL Systems Fast Arithmetic Hardware Smart Cards Practical Attacks on Smart Cards Mathematical Countermeasures Against Side-Channel Attacks Random Numbers-Generation and Testing References

1,113 citations

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TL;DR: A deterministic algorithm to compute the number of F^-points of an elliptic curve that is defined over a finite field Fv and which is given by a Weierstrass equation is presented.

Abstract: In this paper we present a deterministic algorithm to compute the number of F^-points of an elliptic curve that is defined over a finite field Fv and which is given by a Weierstrass equation. The algorithm takes 0(log9 q) elementary operations. As an application wc give an algorithm to compute square roots mod p. For fixed .i e Z, it takes 0(log9p) elementary operations to compute fx mod p. 1. Introduction. In this paper we present an algorithm to compute the number of F(/-points of an elliptic curve defined over a finite field F , which is given by a Weierstrass equation. We restrict ourselves to the case where the characteristic of F^ is not 2 or 3. The algorithm is deterministic, does not depend on any unproved hypotheses and takes 0(log9 0. If one applies fast multiplication techniques, the algorithm will take 0((|x|1/2log p)6+f) elementary operations for any e > 0. Let £ be an elliptic curve defined over the prime field Fp and let an affine model of it be given by a Weierstrass equation Y2 = X3 + AX + B (A,BeFp). An explicit formula for the number of F^-points on £ is given by

674 citations

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28 May 2003

TL;DR: This book discusses Elliptic Curve Cryptography, a Cryptosystem Based on Factoring and its Applications, and some of the techniques used to develop such systems.

Abstract: INTRODUCTION THE BASIC THEORY Weierstrass Equations The Group Law Projective Space and the Point at Infinity Proof of Associativity Other Equations for Elliptic Curves Other Coordinate Systems The j-Invariant Elliptic Curves in Characteristic 2 Endomorphisms Singular Curves Elliptic Curves mod n TORSION POINTS Torsion Points Division Polynomials The Weil Pairing The Tate-Lichtenbaum Pairing Elliptic Curves over Finite Fields Examples The Frobenius Endomorphism Determining the Group Order A Family of Curves Schoof's Algorithm Supersingular Curves The Discrete Logarithm Problem The Index Calculus General Attacks on Discrete Logs Attacks with Pairings Anomalous Curves Other Attacks Elliptic Curve Cryptography The Basic Setup Diffie-Hellman Key Exchange Massey-Omura Encryption ElGamal Public Key Encryption ElGamal Digital Signatures The Digital Signature Algorithm ECIES A Public Key Scheme Based on Factoring A Cryptosystem Based on the Weil Pairing Other Applications Factoring Using Elliptic Curves Primality Testing Elliptic Curves over Q The Torsion Subgroup: The Lutz-Nagell Theorem Descent and the Weak Mordell-Weil Theorem Heights and the Mordell-Weil Theorem Examples The Height Pairing Fermat's Infinite Descent 2-Selmer Groups Shafarevich-Tate Groups A Nontrivial Shafarevich-Tate Group Galois Cohomology Elliptic Curves over C Doubly Periodic Functions Tori Are Elliptic Curves Elliptic Curves over C Computing Periods Division Polynomials The Torsion Subgroup: Doud's Method Complex Multiplication Elliptic Curves over C Elliptic Curves over Finite Fields Integrality of j-Invariants Numerical Examples Kronecker's Jugendtraum DIVISORS Definitions and Examples The Weil Pairing The Tate-Lichtenbaum Pairing Computation of the Pairings Genus One Curves and Elliptic Curves Equivalence of the Definitions of the Pairings Nondegeneracy of the Tate-Lichtenbaum Pairing ISOGENIES The Complex Theory The Algebraic Theory Velu's Formulas Point Counting Complements Hyperelliptic Curves Basic Definitions Divisors Cantor's Algorithm The Discrete Logarithm Problem Zeta Functions Elliptic Curves over Finite Fields Elliptic Curves over Q Fermat's Last Theorem Overview Galois Representations Sketch of Ribet's Proof Sketch of Wiles's Proof APPENDIX A: NUMBER THEORY APPENDIX B: GROUPS APPENDIX C: FIELDS APPENDIX D: COMPUTER packages REFERENCES INDEX Exercises appear at the end of each chapter.

665 citations