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Elliptic Curves And Primality Proving
 Math. Comp
, 1993
"... The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm. ..."
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Cited by 162 (22 self)
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The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm.
Speeding Up The Computations On An Elliptic Curve Using AdditionSubtraction Chains
 Theoretical Informatics and Applications
, 1990
"... We show how to compute x k using multiplications and divisions. We use this method in the context of elliptic curves for which a law exists with the property that division has the same cost as multiplication. Our best algorithm is 11.11% faster than the ordinary binary algorithm and speeds up acco ..."
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Cited by 100 (4 self)
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We show how to compute x k using multiplications and divisions. We use this method in the context of elliptic curves for which a law exists with the property that division has the same cost as multiplication. Our best algorithm is 11.11% faster than the ordinary binary algorithm and speeds up accordingly the factorization and primality testing algorithms using elliptic curves. 1. Introduction. Recent algorithms used in primality testing and integer factorization make use of elliptic curves defined over finite fields or Artinian rings (cf. Section 2). One can define over these sets an abelian law. As a consequence, one can transpose over the corresponding groups all the classical algorithms that were designed over Z/NZ. In particular, one has the analogue of the p \Gamma 1 factorization algorithm of Pollard [29, 5, 20, 22], the Fermatlike primality testing algorithms [1, 14, 21, 26] and the public key cryptosystems based on RSA [30, 17, 19]. The basic operation performed on an elli...
Elliptic Curves with Complex Multiplication and the Conjecture of Birch and SwinnertonDyer
 Birch and SwinnertonDyer, Invent. Math
, 1981
"... A stronger version of (ii) (with no assumption that E have good reduction above p) was proved in [Ru2]. The program to prove (ii) was also begun by Coates and Wiles; it can ? Partially supported by the National Science Foundation. The author also gratefully acknowledges the CIME for its hospitality ..."
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Cited by 22 (0 self)
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A stronger version of (ii) (with no assumption that E have good reduction above p) was proved in [Ru2]. The program to prove (ii) was also begun by Coates and Wiles; it can ? Partially supported by the National Science Foundation. The author also gratefully acknowledges the CIME for its hospitality. ?? current address: Department of Mathematics, Stanford University, Stanford, CA 94305 USA, rubin@math.stanford.edu now be completed thanks to the recent Euler system machinery of Kolyvagin [Ko]. This proof will be given in x12, Corollary 12.13 and Theorem 12.19. The material through x4 is background which was not in the Cetraro lectures but is included here for completeness. In those sections we summarize, generally with references to [Si] instead of proofs, the basic properties of elliptic curves that will be needed later. For more details, including proofs, see Silverman's b
Explicit 4descents on an elliptic curve
 Acta Arith
, 1996
"... Abstract. It is shown that the obvious method of descending from an element of the 2Selmer group of an elliptic curve, E, will indeed give elements of order 1, 2 or 4 in the WeilChatelet group of E. Explicit algorithms for such a method are given. 1. ..."
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Abstract. It is shown that the obvious method of descending from an element of the 2Selmer group of an elliptic curve, E, will indeed give elements of order 1, 2 or 4 in the WeilChatelet group of E. Explicit algorithms for such a method are given. 1.
Building Cyclic Elliptic Curves Modulo Large Primes
 Advances in Cryptology  EUROCRYPT '91, Lecture Notes in Computer Science
, 1987
"... Elliptic curves play an important role in many areas of modern cryptology such as integer factorization and primality proving. Moreover, they can be used in cryptosystems based on discrete logarithms for building oneway permutations. For the latter purpose, it is required to have cyclic elliptic cu ..."
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Cited by 18 (2 self)
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Elliptic curves play an important role in many areas of modern cryptology such as integer factorization and primality proving. Moreover, they can be used in cryptosystems based on discrete logarithms for building oneway permutations. For the latter purpose, it is required to have cyclic elliptic curves over finite fields. The aim of this note is to explain how to construct such curves over a finite field of large prime cardinality, using the ECPP primality proving test of Atkin and Morain. 1 Introduction Elliptic curves prove to be a powerful tool in modern cryptology. Following the original work of H. W. Lenstra, Jr. [18] concerning integer factorization, many researchers have used this new idea to work out primality proving algorithms [8, 14, 2, 4, 22] as well as cryptosystems [21, 16] generalizing those of [12, 1, 9]. Recent work on these topics can be found in [20, 19]. More recently, Kaliski [15] has used elliptic curves in the design of oneway permutations. For this, the autho...
Finding large Selmer rank via an arithmetic theory of local constants
 Annals of Math. http://arxiv.org/abs/math/0512085
"... Abstract. We obtain lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields. Suppose K/k is a quadratic extension of number fields, E is an elliptic curve defined over k, and p is an odd prime. Let K − denote the maximal abelian pextension of K that is unramified ..."
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Cited by 14 (4 self)
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Abstract. We obtain lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields. Suppose K/k is a quadratic extension of number fields, E is an elliptic curve defined over k, and p is an odd prime. Let K − denote the maximal abelian pextension of K that is unramified at all primes where E has bad reduction and that is Galois over k with dihedral Galois group (i.e., the generator c of Gal(K/k) acts on Gal(K − /K) by −1). We prove (under mild hypotheses on p) that if the Zprank of the prop Selmer group Sp(E/K) is odd, then rankZp Sp(E/F) ≥ [F: K] for every finite extension F of K in K −.
Analysis of the Xedni calculus attack
 Design, Codes and Cryptography
, 2000
"... Abstract. The xedni calculus attack on the elliptic curve discrete logarithm problem (ECDLP) involves lifting points from the finite field Fp to the rational numbers Q and then constructing an elliptic curve over Q that passes through them. If the lifted points are linearly dependent, then the ECDLP ..."
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Cited by 12 (2 self)
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Abstract. The xedni calculus attack on the elliptic curve discrete logarithm problem (ECDLP) involves lifting points from the finite field Fp to the rational numbers Q and then constructing an elliptic curve over Q that passes through them. If the lifted points are linearly dependent, then the ECDLP is solved. Our purpose is to analyze the practicality of this algorithm. We find that asymptotically the algorithm is virtually certain to fail, because of an absolute bound on the size of the coefficients of a relation satisfied by the lifted points. Moreover, even for smaller values of p experiments show that the odds against finding a suitable lifting are prohibitively high.
Periodindex problems in WCgroups I: elliptic curves
 J. Number Theory
, 2005
"... Abstract. Let E/K be an elliptic curve defined over a number field, and let p be a prime number such that E(K) has full ptorsion. We show that the order of the ppart of the ShafarevichTate group of E/L is unbounded as L varies over degree p extensions of K. The proof uses O’Neil’s periodindex obs ..."
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Cited by 11 (7 self)
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Abstract. Let E/K be an elliptic curve defined over a number field, and let p be a prime number such that E(K) has full ptorsion. We show that the order of the ppart of the ShafarevichTate group of E/L is unbounded as L varies over degree p extensions of K. The proof uses O’Neil’s periodindex obstruction. We deduce the result from the fact that, under the same hypotheses, there exist infinitely many elements of the WeilChâtelet group of E/K of period p and index p 2. 1.
There are genus one curves of every index over every number field
 J. Reine Angew. Math
"... Abstract. We show that there exist genus one curves of every index over the rational numbers, answering affirmatively a question of Lang and Tate. The proof is “elementary ” in the sense that it does not assume the finiteness of any ShafarevichTate group. On the other hand, using Kolyvagin’s constr ..."
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Cited by 10 (8 self)
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Abstract. We show that there exist genus one curves of every index over the rational numbers, answering affirmatively a question of Lang and Tate. The proof is “elementary ” in the sense that it does not assume the finiteness of any ShafarevichTate group. On the other hand, using Kolyvagin’s construction of a rational elliptic curve whose MordellWeil and ShafarevichTate groups are both trivial, we show that there are infinitely many genus one curves of every index over every number field. 1.
Implementation Of The AtkinGoldwasserKilian Primality Testing Algorithm
 Rapport de Recherche 911, INRIA, Octobre
, 1988
"... . We describe a primality testing algorithm, due essentially to Atkin, that uses elliptic curves over finite fields and the theory of complex multiplication. In particular, we explain how the use of class fields and genus fields can speed up certain phases of the algorithm. We sketch the actual impl ..."
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Cited by 9 (7 self)
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. We describe a primality testing algorithm, due essentially to Atkin, that uses elliptic curves over finite fields and the theory of complex multiplication. In particular, we explain how the use of class fields and genus fields can speed up certain phases of the algorithm. We sketch the actual implementation of this test and its use on testing large primes, the records being two numbers of more than 550 decimal digits. Finally, we give a precise answer to the question of the reliability of our computations, providing a certificate of primality for a prime number. IMPLEMENTATION DU TEST DE PRIMALITE D' ATKIN, GOLDWASSER, ET KILIAN R'esum'e. Nous d'ecrivons un algorithme de primalit'e, principalement du `a Atkin, qui utilise les propri'et'es des courbes elliptiques sur les corps finis et la th'eorie de la multiplication complexe. En particulier, nous expliquons comment l'utilisation du corps de classe et du corps de genre permet d'acc'el'erer les calculs. Nous esquissons l'impl'ementati...