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Factorization of the tenth and eleventh Fermat numbers
, 1996
"... . We describe the complete factorization of the tenth and eleventh Fermat numbers. The tenth Fermat number is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The eleventh Fermat number is a product of five prime factors with 6, 6, 21, 22 and 564 decimal digits. We also note a ..."
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Cited by 17 (8 self)
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. We describe the complete factorization of the tenth and eleventh Fermat numbers. The tenth Fermat number is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The eleventh Fermat number is a product of five prime factors with 6, 6, 21, 22 and 564 decimal digits. We also note a new 27decimal digit factor of the thirteenth Fermat number. This number has four known prime factors and a 2391decimal digit composite factor. All the new factors reported here were found by the elliptic curve method (ECM). The 40digit factor of the tenth Fermat number was found after about 140 Mflopyears of computation. We discuss aspects of the practical implementation of ECM, including the use of specialpurpose hardware, and note several other large factors found recently by ECM. 1. Introduction For a nonnegative integer n, the nth Fermat number is F n = 2 2 n + 1. It is known that F n is prime for 0 n 4, and composite for 5 n 23. Also, for n 2, the factors of F n are of th...
Class Field Theory in Characteristic p, its Origin and Development
 the Proceedings of the International Conference on Class Field Theory (Tokyo
, 2002
"... Today's notion of "global field" comprises number fields (algebraic, of finite degree) and function fields (algebraic, of dimension 1, finite base field). They have many similar arithmetic properties. The systematic study of these similarities seems to have been started by Dedekind (1857). A new imp ..."
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Cited by 14 (5 self)
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Today's notion of "global field" comprises number fields (algebraic, of finite degree) and function fields (algebraic, of dimension 1, finite base field). They have many similar arithmetic properties. The systematic study of these similarities seems to have been started by Dedekind (1857). A new impetus was given by the seminal thesis of E. Artin (1921, published in 1924). In this exposition I shall report on the development during the twenties and thirties of our century, with emphasis on the emergence of class field theory for function elds. The names of F.K.Schmidt, H. Hasse, E. Witt, C. Chevalley (among others) are closely connected with that development.
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...
Finding Meaning in Error Terms
, 2007
"... (In memory of Serge Lang) Four decades ago, Mikio Sato and John Tate predicted the shape of probability distributions to which certain “error terms ” in number theory conform. Their prediction—known as the SatoTate ..."
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Cited by 9 (1 self)
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(In memory of Serge Lang) Four decades ago, Mikio Sato and John Tate predicted the shape of probability distributions to which certain “error terms ” in number theory conform. Their prediction—known as the SatoTate
Finding Secure Curves with the SatohFGH Algorithm and an EarlyAbort Strategy
 in B. P (ed), Advances in Cryptology  EUROCRYPT 2001, Lecture Notes in Computer Science 2045
, 2001
"... The use of elliptic curves in cryptography relies on the ability to count the number of points on a given curve. Before 1999, the SEA algorithm was the only ecient method known for random curves. Then Satoh proposed a new algorithm based on the canonical padic lift of the curve for p 5. In an ..."
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Cited by 5 (2 self)
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The use of elliptic curves in cryptography relies on the ability to count the number of points on a given curve. Before 1999, the SEA algorithm was the only ecient method known for random curves. Then Satoh proposed a new algorithm based on the canonical padic lift of the curve for p 5. In an earlier paper, the authors extended Satoh's method to the case of characteristics two and three. This paper presents an implementation of the SatohFGH algorithm and its application to the problem of nding curves suitable for cryptography. By combining SatohFGH and an earlyabort strategy based on SEA, we are able to nd secure random curves in characteristic two in much less time than previously reported. In particular we can generate curves widely considered to be as secure as RSA1024 in less than one minute each on a fast workstation.
Basic Algorithms for Elliptic Curves
"... b 2 b 4 216b 6 : The discriminant is then = b 2 2 b 8 8b 3 4 27b 2 6 + 9b 2 b 4 b 6 and the modular invariant j = c 3 4 : However, if E is dened over a number eld K, we shall use the short Weierstrass form (1:2) E : Y 2 = X 3 + aX + b (a; b 2 K) with discriminant = 16(4a 3 + 27b ..."
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b 2 b 4 216b 6 : The discriminant is then = b 2 2 b 8 8b 3 4 27b 2 6 + 9b 2 b 4 b 6 and the modular invariant j = c 3 4 : However, if E is dened over a number eld K, we shall use the short Weierstrass form (1:2) E : Y 2 = X 3 + aX + b (a; b 2 K) with discriminant = 16(4a 3 + 27b 2 ) = 16 0 and modular invariant j = 12 3 4a 3<F12.2
my sponsorbody over the last eight years.
, 2006
"... I would like to thank my supervisor Peter Brown for his time and devotion throughout this year. Without his useful suggestions (for both mathematics and English), this thesis certainly could not have come this far. I also wish to thank all of my fellow Honours students for their direct and indirect ..."
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I would like to thank my supervisor Peter Brown for his time and devotion throughout this year. Without his useful suggestions (for both mathematics and English), this thesis certainly could not have come this far. I also wish to thank all of my fellow Honours students for their direct and indirect support during the year. Their busy lifestyles always make me ashamed, but encourage me to take my responsibility seriously. My family also deserves some thanks for their support during my (hard) time overseas. Although they might not have much idea what sort of science I am majoring in, I hope to make them aware of it in a very near future.
A NOTE ON ZEROS OF LSERIES OF ELLIPTIC CURVES
, 2006
"... Abstract. In this note we study an analogy between a positive definite quadratic form for elliptic curves over finite fields and a positive definite quadratic form for elliptic curves over the rational number field. A question is posed of which an affirmative answer would imply the analogue of the R ..."
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Abstract. In this note we study an analogy between a positive definite quadratic form for elliptic curves over finite fields and a positive definite quadratic form for elliptic curves over the rational number field. A question is posed of which an affirmative answer would imply the analogue of the Riemann hypothesis for elliptic curves over the rational number field.
Emergent Spacetime from Modular Motives
, 812
"... The program of constructing spacetime geometry from string theoretic modular forms is extended to CalabiYau varieties of dimensions two, three, and four, as well as higher rank motives. Modular forms on the worldsheet can be constructed from the geometry of spacetime by computing the Lfunctions as ..."
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The program of constructing spacetime geometry from string theoretic modular forms is extended to CalabiYau varieties of dimensions two, three, and four, as well as higher rank motives. Modular forms on the worldsheet can be constructed from the geometry of spacetime by computing the Lfunctions associated to omega motives of CalabiYau varieties, generated by their holomorphic n−forms via Galois representations. The modular forms that emerge from the Ω−motive and other motives of the intermediate cohomology are related to characters of the underlying rational conformal field theory. The converse problem of constructing space from string theory proceeds in the class of diagonal theories by determining the motives associated to modular forms in the category of pure motives with complex multiplication. The emerging picture indicates that the Lfunction can be interpreted