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124
Algorithms for computing isogenies between elliptic curves
 Math. Comp
, 2000
"... Abstract. The heart of the improvements by Elkies to Schoof’s algorithm for computing the cardinality of elliptic curves over a finite field is the ability to compute isogenies between curves. Elkies ’ approach is well suited for the case where the characteristic of the field is large. Couveignes sh ..."
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Cited by 31 (6 self)
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Abstract. The heart of the improvements by Elkies to Schoof’s algorithm for computing the cardinality of elliptic curves over a finite field is the ability to compute isogenies between curves. Elkies ’ approach is well suited for the case where the characteristic of the field is large. Couveignes showed how to compute isogenies in small characteristic. The aim of this paper is to describe the first successful implementation of Couveignes’s algorithm. In particular, we describe the use of fast algorithms for performing incremental operations on series. We also insist on the particular case of the characteristic 2. 1.
Finding Suitable Curves For The Elliptic Curve Method Of Factorization
 Math. Comp
, 1993
"... Using the parametrizations of Kubert, we show how to produce infinite families of elliptic curves which have prescribed nontrivial torsion over Q and rank at least one. These curves can be used to speed up the ECM factorization algorithm of Lenstra. We also briefly discuss curves with complex multip ..."
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Cited by 30 (2 self)
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Using the parametrizations of Kubert, we show how to produce infinite families of elliptic curves which have prescribed nontrivial torsion over Q and rank at least one. These curves can be used to speed up the ECM factorization algorithm of Lenstra. We also briefly discuss curves with complex multiplication in this context. 1 Introduction 1.1 The ECM method of Lenstra [5] for finding a prime factor p of a number N uses a "random" elliptic curve E : y 2 = f(x) = x 3 + ax + b: If the number k of points on E modulo p is smooth, the method succeeds. Suyama [9] and Montgomery [7] developed infinite classes of curves E for which k has some prescribed small factors; on reasonable probabilistic assumptions (borne out in practice) this should lead to a slight improvement in the method. Specifically, Montgomery and Suyama each force a factor of 12 in k, and Montgomery forces a factor of 16 but only on the assumption that p is congruent to 1 modulo 4. In this paper, we show how to force a...
Constructing hyperelliptic curves of genus 2 suitable for cryptography
 Math. Comp
, 2003
"... Abstract. In this article we show how to generalize the CMmethod for elliptic curves to genus two. We describe the algorithm in detail and discuss the results of our implementation. 1. ..."
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Cited by 29 (2 self)
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Abstract. In this article we show how to generalize the CMmethod for elliptic curves to genus two. We describe the algorithm in detail and discuss the results of our implementation. 1.
Schoof's Algorithm and Isogeny Cycles
, 1994
"... . The heart of Schoof's algorithm for computing the cardinality m of an elliptic curve over a finite field is the computation of m modulo small primes `. Elkies and Atkin have designed practical improvements to the basic algorithm, that make use of "good" primes `. We show how to use powers of go ..."
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Cited by 28 (6 self)
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. The heart of Schoof's algorithm for computing the cardinality m of an elliptic curve over a finite field is the computation of m modulo small primes `. Elkies and Atkin have designed practical improvements to the basic algorithm, that make use of "good" primes `. We show how to use powers of good primes in an efficient way. This is done by computing isogenies between curves over the ground field. A new structure appears, called "isogeny cycle". We investigate some properties of this structure. 1 Introduction Let E be an elliptic curve over a primitive finite field F p where p is a large prime integer. (We are not dealing with the case of small characteristic here.) The curve is given by some equation E(X; Y ) = 0 in Weierstrass form E(X; Y ) = Y 2 \Gamma X 3 \Gamma AX \Gamma B so that a generic point on the curve is given by (X; Y ) mod E . Let m be the number of points of E. It is well known that m = p + 1 \Gamma t, with t an integer satisfying jtj ! 2 p p. If p is small...
Implementing the asymptotically fast version of the elliptic curve primality proving algorithm
 Math. Comp
"... Abstract. The elliptic curve primality proving (ECPP) algorithm is one of the current fastest practical algorithms for proving the primality of large numbers. Its running time cannot be proven rigorously, but heuristic arguments show that it should run in time Õ((log N)5) to prove the primality of N ..."
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Cited by 27 (1 self)
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Abstract. The elliptic curve primality proving (ECPP) algorithm is one of the current fastest practical algorithms for proving the primality of large numbers. Its running time cannot be proven rigorously, but heuristic arguments show that it should run in time Õ((log N)5) to prove the primality of N. An asymptotically fast version of it, attributed to J. O. Shallit, runs in time Õ((log N)4). The aim of this article is to describe this version in more details, leading to actual implementations able to handle numbers with several thousands of decimal digits. 1.
Action of modular correspondences around CM points
"... We study the action of modular correspondences in the p adic neighborhood of CM points. We deduce and prove two stable and ecient padic analytic methods for computing singular values of modular functions. On the way we prove a non trivial lower bound for the density of smooth numbers in imagin ..."
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Cited by 25 (0 self)
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We study the action of modular correspondences in the p adic neighborhood of CM points. We deduce and prove two stable and ecient padic analytic methods for computing singular values of modular functions. On the way we prove a non trivial lower bound for the density of smooth numbers in imaginary quadratic rings and show that the canonical lift of an elliptic curve over Fq can be computed in probabilistic time exp((log q) ) under GRH. We also extend the notion of canonical lift to supersingular elliptic curves and show how to compute it in that case.
Primality testing using elliptic curves
 Journal of the ACM
, 1999
"... Abstract. We present a primality proving algorithm—a probabilistic primality test that produces short certificates of primality on prime inputs. We prove that the test runs in expected polynomial time for all but a vanishingly small fraction of the primes. As a corollary, we obtain an algorithm for ..."
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Cited by 22 (0 self)
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Abstract. We present a primality proving algorithm—a probabilistic primality test that produces short certificates of primality on prime inputs. We prove that the test runs in expected polynomial time for all but a vanishingly small fraction of the primes. As a corollary, we obtain an algorithm for generating large certified primes with distribution statistically close to uniform. Under the conjecture that the gap between consecutive primes is bounded by some polynomial in their size, the test is shown to run in expected polynomial time for all primes, yielding a Las Vegas primality test. Our test is based on a new methodology for applying group theory to the problem of prime certification, and the application of this methodology using groups generated by elliptic curves over finite fields. We note that our methodology and methods have been subsequently used and improved upon, most notably in the primality proving algorithm of Adleman and Huang using hyperelliptic curves and
Factorization Of The Tenth Fermat Number
 MATH. COMP
, 1999
"... We describe the complete factorization of the tenth Fermat number F 10 by the elliptic curve method (ECM). F 10 is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The 40digit factor was found after about 140 Mflopyears of computation. We also discuss the complete factor ..."
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Cited by 21 (10 self)
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We describe the complete factorization of the tenth Fermat number F 10 by the elliptic curve method (ECM). F 10 is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The 40digit factor was found after about 140 Mflopyears of computation. We also discuss the complete factorization of other Fermat numbers by ECM, and summarize the factorizations of F 5 ; : : : ; F 11 .
A CRT algorithm for constructing genus 2 curves over finite fields
, 2007
"... Abstract. — We present a new method for constructing genus 2 curves over a finite field Fn with a given number of points on its Jacobian. This method has important applications in cryptography, where groups of prime order are used as the basis for discretelog based cryptosystems. Our algorithm prov ..."
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Cited by 19 (7 self)
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Abstract. — We present a new method for constructing genus 2 curves over a finite field Fn with a given number of points on its Jacobian. This method has important applications in cryptography, where groups of prime order are used as the basis for discretelog based cryptosystems. Our algorithm provides an alternative to the traditional CM method for constructing genus 2 curves. For a quartic CM field K with primitive CM type, we compute the Igusa class polynomials modulo p for certain small primes p and then use the Chinese remainder theorem (CRT) and a bound on the denominators to construct the class polynomials. We also provide an algorithm for determining endomorphism rings of ordinary Jacobians of genus 2 curves over finite fields, generalizing the work of Kohel for elliptic curves. Résumé (Un algorithme fondé sur le théorème chinois pour construire des courbes de genre 2 sur des corps finis) Nous présentons une nouvelle méthode pour construire des courbes de genre 2 sur un corps fini Fn avec un nombre donné de points sur sa jacobienne. Cette méthode a des applications importantes en cryptographie, où des groupes d’ordre premier sont employés pour former des cryptosystèmes fondés sur le logarithme discret. Notre algorithme fournit une alternative à la méthode traditionnelle de multiplication complexe pour construire des courbes de genre 2. Pour un corps quartique K à multiplication complexe de type primitif, nous calculons les polynômes de classe d’Igusa modulo p pour certain petit premiers p et employons le théorème chinois et une borne sur les dénominateurs pour construire les polynômes de classe. Nous fournissons également un algorithme pour déterminer les anneaux d’endomorphismes des jacobiennes de courbes ordinaires de genre 2 sur des corps finis, généralisant le travail de Kohel pour les courbes elliptiques.
Constructing elliptic curves with a known number of points over a prime field
 Primes and Misdemeanours: Lectures in Honour of the 60th Birthday of H C Williams, volume 41 of Fields Inst. Commun
, 2004
"... Abstract. Elliptic curves with a known number of points over a given prime field Fn are often needed for use in cryptography. In the context of primality proving, Atkin and Morain suggested the use of the theory of complex multiplication to construct such curves. One of the steps in this method is t ..."
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Cited by 18 (8 self)
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Abstract. Elliptic curves with a known number of points over a given prime field Fn are often needed for use in cryptography. In the context of primality proving, Atkin and Morain suggested the use of the theory of complex multiplication to construct such curves. One of the steps in this method is the calculation of a root modulo n of the Hilbert class polynomial HD(X) for a fundamental discriminant D. The usual way is to compute HD(X) over the integers and then to find the root modulo n. We present a modified version of the Chinese remainder theorem (CRT) to compute HD(X) modulo n directly from the knowledge of HD(X) modulo enough small primes. Our complexity analysis suggests that asymptotically our algorithm is an improvement over previously known methods. 1.