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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.
Explicit Bounds on Exponential Sums and the Scarcity of Squarefree Binomial Coefficients
, 1996
"... This paper fills what we believe to be a lacuna in the existing literature concerning upper bounds on exponential sums. Although it has always been evident that many of the known estimates can be made explicit, it is a nontrivial problem to actually do so. In particular so that the constants involv ..."
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Cited by 15 (1 self)
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This paper fills what we believe to be a lacuna in the existing literature concerning upper bounds on exponential sums. Although it has always been evident that many of the known estimates can be made explicit, it is a nontrivial problem to actually do so. In particular so that the constants involved do not render the explicit estimates useless in practical applications. We have used the practical bounds that are needed to prove Theorem 1 as motivation for our results here, though we hope that this work will be applicable to a variety of other problems which routinely apply these or related exponential sum estimates. In particular our results here can be used to say something about the questions of estimating the number of integers free of large prime factors in short intervals (see [FL]), and of the largest prime factor of an integer in an interval (see [J]). Our key result is
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...
Elliptic Curves, Primality Proving And Some Titanic Primes
, 1989
"... We describe how to generate large primes using the primality proving algorithm of Atkin. Figure 1: The Titanic . 1. Introduction. During the last ten years, primality testing evolved at great speed. Motivated by the RSA cryptosystem [3], the first deterministic primality proving algorithm was de ..."
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Cited by 5 (3 self)
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We describe how to generate large primes using the primality proving algorithm of Atkin. Figure 1: The Titanic . 1. Introduction. During the last ten years, primality testing evolved at great speed. Motivated by the RSA cryptosystem [3], the first deterministic primality proving algorithm was designed by Adleman, Pomerance and Rumely [2] and made practical by Cohen, H. W. Lenstra and A. K. Lenstra (see [9, 10] and more recently [5]). It was then proved that the time needed to test an arbitrary integer N for primality is O((log N) c log log log N ) for some positive constant c ? 0. When implemented on a huge computer, the algorithm was able to test 200 digit numbers in about 10 minutes of CPU time. A few years ago, Goldwasser and Kilian [11], used the theory of elliptic curves over finite fields to give the first primality proving algorithm whose running time is polynomial in log N (assuming a plausible conjecture in number theory). Atkin [4] used the theory of complex multiplicat...
A Problem Concerning a Character Sum
, 1999
"... this paper we exhibit some techniques which were successful in producing, for each k such that 3 k 2000, a value for p such that S(k) > 0. 1. INTRODUCTION ..."
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Cited by 4 (1 self)
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this paper we exhibit some techniques which were successful in producing, for each k such that 3 k 2000, a value for p such that S(k) > 0. 1. INTRODUCTION
Building Pseudoprimes With A Large Number Of Prime Factors
, 1995
"... We extend the method due originally to Loh and Niebuhr for the generation of Carmichael numbers with a large number of prime factors to other classes of pseudoprimes, such as Williams's pseudoprimes and elliptic pseudoprimes. We exhibit also some new Dickson pseudoprimes as well as superstrong Dicks ..."
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We extend the method due originally to Loh and Niebuhr for the generation of Carmichael numbers with a large number of prime factors to other classes of pseudoprimes, such as Williams's pseudoprimes and elliptic pseudoprimes. We exhibit also some new Dickson pseudoprimes as well as superstrong Dickson pseudoprimes.
Probabilistic Primality Testing
 Analysis of Algorithms Seminar I. INRIA Research Report XXX
, 1992
"... Introduction The prototype of pseudoprime tests is Fermat's theorem: If p is a prime and a an integer prime to p, then a p\Gamma1 j 1 mod p: A pseudoprime to base a (pspa) is a composite number N such that a N \Gamma1 j 1 mod N: For all a, there exists an infinite number of pspa. Moreove ..."
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Introduction The prototype of pseudoprime tests is Fermat's theorem: If p is a prime and a an integer prime to p, then a p\Gamma1 j 1 mod p: A pseudoprime to base a (pspa) is a composite number N such that a N \Gamma1 j 1 mod N: For all a, there exists an infinite number of pspa. Moreover, there are numbers N such that N is a pspa for all a. (These numbers are called Carmichael numbers.) A refinement of this test consists in writing N = 1 +N 0 2 t with N 0 odd and a N \Gamma1 \Gamma 1 = (a N 0<F12.96
Vinogradov's Theorem Is True Up To 10^20
, 1995
"... : Vinogradov's theorem states that any sufficiently odd integer is the sum of three prime numbers. This theorem allows to suppose the conjecture that this is true for all odd integers. In this paper, we describe the implementation of an algorithm which allowed to check this conjecture up to 10 20 ..."
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: Vinogradov's theorem states that any sufficiently odd integer is the sum of three prime numbers. This theorem allows to suppose the conjecture that this is true for all odd integers. In this paper, we describe the implementation of an algorithm which allowed to check this conjecture up to 10 20 . Keywords: Prime numbers; Goldbach's conjecture; Vinogradov's theorem (R'esum'e : tsvp) y email: saouter@irisa.fr CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE Centre National de la Recherche Scientifique Institut National de Recherche en Informatique (URA 227) Universite de Rennes 1  Insa de Rennes et en Automatique  unite de recherche de Rennes Le th'eor`eme de Vinogradov est vrai jusqu'`a 10 20 R'esum'e : Le th'eor`eme de Vinogradov d'emontre que tout entier impair suffisamment grand est la somme de trois nombres premiers. Ce th'eor`eme permet de supposer la conjecture que cela soit vrai pour tout entier impair. Dans cet article, nous pr'esentons la mise en oeuvre d'un algor...
Easy numbers for the Elliptic Curve Primality Proving Algorithm
, 1992
"... We present some new classes of numbers that are easier to test for primality with the Elliptic Curve Primality Proving algorithm than average numbers. It is shown that this is the case for about half the numbers of the Cunningham project. Computational examples are given. ..."
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We present some new classes of numbers that are easier to test for primality with the Elliptic Curve Primality Proving algorithm than average numbers. It is shown that this is the case for about half the numbers of the Cunningham project. Computational examples are given.