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Guide to Elliptic Curve Cryptography
, 2004
"... Elliptic curves have been intensively studied in number theory and algebraic geometry for over 100 years and there is an enormous amount of literature on the subject. To quote the mathematician Serge Lang: It is possible to write endlessly on elliptic curves. (This is not a threat.) Elliptic curves ..."
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Cited by 593 (18 self)
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Elliptic curves have been intensively studied in number theory and algebraic geometry for over 100 years and there is an enormous amount of literature on the subject. To quote the mathematician Serge Lang: It is possible to write endlessly on elliptic curves. (This is not a threat.) Elliptic curves
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 201 (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.
Modular elliptic curves and Fermat’s Last Theorem
 ANNALS OF MATH
, 1995
"... When Andrew John Wiles was 10 years old, he read Eric Temple Bell’s The Last Problem and was so impressed by it that he decided that he would be the first person to prove Fermat’s Last Theorem. This theorem states that there are no nonzero integers a, b, c, n with n> 2 such that a n + b n = c n ..."
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Cited by 612 (1 self)
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n. The object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular. Fermat’s Last Theorem follows as a corollary by virtue of previous work by Frey, Serre and Ribet.
Vector bundles over an elliptic curve
 Proc. London Math. Soc
, 1957
"... THE primary purpose of this paper is the study of algebraic vector bundles over an elliptic curve (defined over an algebraically closed field k). The interest of the elliptic curve lies in the fact that it provides the first nontrivial case, Grothendieck (6) having shown that for a rational curve e ..."
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Cited by 301 (0 self)
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THE primary purpose of this paper is the study of algebraic vector bundles over an elliptic curve (defined over an algebraically closed field k). The interest of the elliptic curve lies in the fact that it provides the first nontrivial case, Grothendieck (6) having shown that for a rational curve
Elliptic Curves
, 1996
"... . These are the notes for Math 679, University of Michigan, Winter 1996, exactly as they were handed out during the course except for some minor corrections. Please send comments and corrections to me at jmilne@umich.edu using "Math679" as the subject. Contents Introduction 1 Fast factor ..."
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Cited by 6 (2 self)
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on Algebraic Curves and the RiemannRoch Theorem 14 Regular functions on affine curves Regular functions on projective curves The RiemannRoch theorem The group law revisited Perfect base fields 5. Definition of an Elliptic Curve 19 Plane projective cubic curves with a rational inflection point General
ARITHMETIC OF ELLIPTIC CURVES Elliptic Curves
, 2015
"... ellipticcurve groups (over finite fields) in cryptosystems. Use of supersingular curves discarded after the proposal of the Menezes–Okamoto–Vanstone (1993) or Frey–Rück (1994) attack. ECDSA was proposed by Johnson and Menezes (1999) and adopted as a digital signature standard. Use of pairing in ne ..."
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ellipticcurve groups (over finite fields) in cryptosystems. Use of supersingular curves discarded after the proposal of the Menezes–Okamoto–Vanstone (1993) or Frey–Rück (1994) attack. ECDSA was proposed by Johnson and Menezes (1999) and adopted as a digital signature standard. Use of pairing
Elliptic Curves
 in [Buhler and Stevenhagen 2007]. Citations in this document: §4
"... . This is a introduction to some aspects of the arithmetic of elliptic curves, intended for readers with little or no background in number theory and algebraic geometry. In keeping with the rest of this volume, the presentation has an algorithmic slant. We also touch lightly on curves of higher genu ..."
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Cited by 1 (0 self)
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. This is a introduction to some aspects of the arithmetic of elliptic curves, intended for readers with little or no background in number theory and algebraic geometry. In keeping with the rest of this volume, the presentation has an algorithmic slant. We also touch lightly on curves of higher
Curves for the Elliptic Curves Cryptosystem
"... We use two methods to search for curves for the elliptic curve cryptosystem. The first method involves the definition of an elliptic curve over a number field and its reduction modulo prime ideals. The second method defines an elliptic curve over a small finite field and then considers it over exten ..."
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We use two methods to search for curves for the elliptic curve cryptosystem. The first method involves the definition of an elliptic curve over a number field and its reduction modulo prime ideals. The second method defines an elliptic curve over a small finite field and then considers it over
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 120 (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
Results 1  10
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135,459