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33
Rational Points on Elliptic Curves
, 1992
"... Abstract. We give a quantitative bound for the number of Sintegral points on an elliptic curve over a number field K in terms of the number of primes dividing the denominator of the jinvariant, the degree [K: Q], and the number of primes in S. Let K be a number field of degree d and MK the set of ..."
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Cited by 71 (1 self)
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Abstract. We give a quantitative bound for the number of Sintegral points on an elliptic curve over a number field K in terms of the number of primes dividing the denominator of the jinvariant, the degree [K: Q], and the number of primes in S. Let K be a number field of degree d and MK the set of places of K. Let E/K be an elliptic curve with quasiminimal Weierstrass equation E: y 2 = x 3 + Ax + B. If ∆ = 4A 3 + 27B 2 is the discriminant of this equation, recall that quasiminimal means that N K/Q(∆)  is minimized subject to the condition that A, B ∈ OK. Let S ⊂ MK be a finite set of s places containing all the archimedean ones, and denote the ring of Sintegers by OS. Let j be the jinvariant of E. In [Sil6], Silverman proved that if j is integral, then #{P ∈ E(K) : x(P) ∈ OS} can be bounded in terms of the field K, #S, and the rank of E(K). More generally, Silverman proved that if the jinvariant is nonintegral for at most δ places of K, then that set can be bounded in terms of the previously mentioned constants and δ. This is a special case of a conjecture of Lang asserting the existence of such a bound which is independent of δ. However, Silverman did not explicitly compute the constants involved. In this paper, using more explicit methods, we compute the dependence of the bounds on the various constants. In particular, as a consequence of Proposition 11, we have the following Theorem. For elliptic curves E/K of sufficiently large height, the number of Sintegral points is at most 2 · 10 11 dδ(j) 3d (32 · 10 9) rδ(j)+s. For elliptic curves E defined over Q of sufficiently large height, the number of Sintegral points is at most 32 · 10 11 (32 · 10 9) rδ(j)+s.
Some integer factorization algorithms using elliptic curves
 Australian Computer Science Communications
, 1986
"... Lenstra’s integer factorization algorithm is asymptotically one of the fastest known algorithms, and is also ideally suited for parallel computation. We suggest a way in which the algorithm can be speeded up by the addition of a second phase. Under some plausible assumptions, the speedup is of order ..."
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Cited by 47 (13 self)
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Lenstra’s integer factorization algorithm is asymptotically one of the fastest known algorithms, and is also ideally suited for parallel computation. We suggest a way in which the algorithm can be speeded up by the addition of a second phase. Under some plausible assumptions, the speedup is of order log(p), where p is the factor which is found. In practice the speedup is significant. We mention some refinements which give greater speedup, an alternative way of implementing a second phase, and the connection with Pollard’s “p − 1” factorization algorithm. 1
Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms
 ACTA ARITHMETICA
, 1994
"... ..."
Parallel Algorithms for Integer Factorisation
"... The problem of finding the prime factors of large composite numbers has always been of mathematical interest. With the advent of public key cryptosystems it is also of practical importance, because the security of some of these cryptosystems, such as the RivestShamirAdelman (RSA) system, depends o ..."
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Cited by 41 (17 self)
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The problem of finding the prime factors of large composite numbers has always been of mathematical interest. With the advent of public key cryptosystems it is also of practical importance, because the security of some of these cryptosystems, such as the RivestShamirAdelman (RSA) system, depends on the difficulty of factoring the public keys. In recent years the best known integer factorisation algorithms have improved greatly, to the point where it is now easy to factor a 60decimal digit number, and possible to factor numbers larger than 120 decimal digits, given the availability of enough computing power. We describe several algorithms, including the elliptic curve method (ECM), and the multiplepolynomial quadratic sieve (MPQS) algorithm, and discuss their parallel implementation. It turns out that some of the algorithms are very well suited to parallel implementation. Doubling the degree of parallelism (i.e. the amount of hardware devoted to the problem) roughly increases the size of a number which can be factored in a fixed time by 3 decimal digits. Some recent computational results are mentioned – for example, the complete factorisation of the 617decimal digit Fermat number F11 = 2211 + 1 which was accomplished using ECM.
Symmetries of Polynomials
 J. Symb. Comp
"... New algorithms for determining discrete and continuous symmetries of polynomials  also known as binary forms in classical invariant theory  are presented. Implementations in Mathematica and Maple are discussed and compared. The results are based on a new, comprehensive theory of moving frames ..."
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Cited by 21 (15 self)
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New algorithms for determining discrete and continuous symmetries of polynomials  also known as binary forms in classical invariant theory  are presented. Implementations in Mathematica and Maple are discussed and compared. The results are based on a new, comprehensive theory of moving frames that completely characterizes the equivalence and symmetry properties of submanifolds under general Lie group actions. This work was partially supported by NSF Grant DMS 9803154. 1 Introduction. The purpose of this paper is to explain the detailed implementation of a new algorithm for determining the symmetries of polynomials (binary forms). The method was first described in the second author's new book [24], and the present paper adds details and refinements. We shall demonstrate that the symmetry group of both real and complex binary forms can be completely determined by solving two simultaneous bivariate polynomial equations, which are based on two fundamental covariants of the for...
Recent progress and prospects for integer factorisation algorithms
 In Proc. of COCOON 2000
, 2000
"... Abstract. The integer factorisation and discrete logarithm problems are of practical importance because of the widespread use of public key cryptosystems whose security depends on the presumed difficulty of solving these problems. This paper considers primarily the integer factorisation problem. In ..."
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Cited by 20 (1 self)
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Abstract. The integer factorisation and discrete logarithm problems are of practical importance because of the widespread use of public key cryptosystems whose security depends on the presumed difficulty of solving these problems. This paper considers primarily the integer factorisation problem. In recent years the limits of the best integer factorisation algorithms have been extended greatly, due in part to Moore’s law and in part to algorithmic improvements. It is now routine to factor 100decimal digit numbers, and feasible to factor numbers of 155 decimal digits (512 bits). We outline several integer factorisation algorithms, consider their suitability for implementation on parallel machines, and give examples of their current capabilities. In particular, we consider the problem of parallel solution of the large, sparse linear systems which arise with the MPQS and NFS methods. 1
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.
Integral points on elliptic curves and 3torsion in class groups
"... We give new bounds for the number of integral points on elliptic curves. The method may be said to interpolate between approaches via diophantine techniques ([BP], [HBR]) and methods based on quasiorthogonality in the MordellWeil lattice ([Sil6], [GS], [He]). We apply our results to break previous ..."
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Cited by 11 (3 self)
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We give new bounds for the number of integral points on elliptic curves. The method may be said to interpolate between approaches via diophantine techniques ([BP], [HBR]) and methods based on quasiorthogonality in the MordellWeil lattice ([Sil6], [GS], [He]). We apply our results to break previous
Open Diophantine Problems
 MOSCOW MATHEMATICAL JOURNAL
, 2004
"... Diophantine Analysis is a very active domain of mathematical research where one finds more conjectures than results. We collect here a number of open questions concerning Diophantine equations (including Pillai’s Conjectures), Diophantine approximation (featuring the abc Conjecture) and transcendent ..."
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Cited by 10 (3 self)
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Diophantine Analysis is a very active domain of mathematical research where one finds more conjectures than results. We collect here a number of open questions concerning Diophantine equations (including Pillai’s Conjectures), Diophantine approximation (featuring the abc Conjecture) and transcendental number theory (with, for instance, Schanuel’s Conjecture). Some questions related to Mahler’s measure and Weil absolute logarithmic height are then considered (e. g., Lehmer’s Problem). We also discuss Mazur’s question regarding the density of rational points on a variety, especially in the particular case of algebraic groups, in connexion with transcendence problems in several variables. We say only a few words on metric problems, equidistribution questions, Diophantine approximation on manifolds and Diophantine analysis on function fields.
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...