Results 1  10
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33
Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms
 ACTA ARITHMETICA
, 1994
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The 2adic CM method for genus 2 curves with application to cryptography
 in ASIACRYPT ‘06, Springer LNCS 4284
, 2006
"... Abstract. The complex multiplication (CM) method for genus 2 is currently the most efficient way of generating genus 2 hyperelliptic curves defined over large prime fields and suitable for cryptography. Since low class number might be seen as a potential threat, it is of interest to push the method ..."
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Cited by 18 (1 self)
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Abstract. The complex multiplication (CM) method for genus 2 is currently the most efficient way of generating genus 2 hyperelliptic curves defined over large prime fields and suitable for cryptography. Since low class number might be seen as a potential threat, it is of interest to push the method as far as possible. We have thus designed a new algorithm for the construction of CM invariants of genus 2 curves, using 2adic lifting of an input curve over a small finite field. This provides a numerically stable alternative to the complex analytic method in the first phase of the CM method for genus 2. As an example we compute an irreducible factor of the Igusa class polynomial system for the quartic CM field Q(i p 75 + 12 √ 17), whose class number is 50. We also introduce a new representation to describe the CM curves: a set of polynomials in (j1, j2, j3) which vanish on the precise set of triples which are the Igusa invariants of curves whose Jacobians have CM by a prescribed field. The new representation provides a speedup in the second phase, which uses Mestre’s algorithm to construct a genus 2 Jacobian of prime order over a large prime field for use in cryptography. 1
Computational Aspects of Curves of Genus at Least 2
 Algorithmic number theory. 5th international symposium. ANTSII
, 1996
"... . This survey discusses algorithms and explicit calculations for curves of genus at least 2 and their Jacobians, mainly over number fields and finite fields. Miscellaneous examples and a list of possible future projects are given at the end. 1. Introduction An enormous number of people have per ..."
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Cited by 14 (3 self)
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. This survey discusses algorithms and explicit calculations for curves of genus at least 2 and their Jacobians, mainly over number fields and finite fields. Miscellaneous examples and a list of possible future projects are given at the end. 1. Introduction An enormous number of people have performed an enormous number of computations on elliptic curves, as one can see from even a perfunctory glance at [29]. A few years ago, the same could not be said for curves of higher genus, even though the theory of such curves had been developed in detail. Now, however, polynomialtime algorithms and sometimes actual programs are available for solving a wide variety of problems associated with such curves. The genus 2 case especially is becoming accessible: in light of recent work, it seems reasonable to expect that within a few years, packages will be available for doing genus 2 computations analogous to the elliptic curve computations that are currently possible in PARI, MAGMA, SIMATH, apec...
The Hasse principle and the BrauerManin obstruction for curves
 Manuscripta Math
, 2004
"... Abstract. We discuss a range of ways, extending existing methods, to demonstrate violations of the Hasse principle on curves. Of particular interest are curves which contain a rational divisor class of degree 1, even though they contain no rational point. For such curves we construct new types of ex ..."
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Abstract. We discuss a range of ways, extending existing methods, to demonstrate violations of the Hasse principle on curves. Of particular interest are curves which contain a rational divisor class of degree 1, even though they contain no rational point. For such curves we construct new types of examples of violations of the Hasse principle which are due to the BrauerManin obstruction, subject to the conjecture that the TateShafarevich group of the Jacobian is finite. 1.
Isogenies and the discrete logarithm problem in jacobians of genus 3 hyperelliptic curves
 Advances in Cryptology — EUROCRYPT 2008, Lecture Notes in Computer Science
"... Abstract. We describe the use of explicit isogenies to reduce Discrete Logarithm Problems (DLPs) on Jacobians of hyperelliptic genus 3 curves to Jacobians of nonhyperelliptic genus 3 curves, which are vulnerable to faster index calculus attacks. We provide algorithms which compute an isogeny with k ..."
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Cited by 11 (2 self)
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Abstract. We describe the use of explicit isogenies to reduce Discrete Logarithm Problems (DLPs) on Jacobians of hyperelliptic genus 3 curves to Jacobians of nonhyperelliptic genus 3 curves, which are vulnerable to faster index calculus attacks. We provide algorithms which compute an isogeny with kernel isomorphic to (Z/2Z) 3 for any hyperelliptic genus 3 curve. These algorithms provide a rational isogeny for a positive fraction of all hyperelliptic genus 3 curves defined over a finite field of characteristic p> 3. Subject to reasonable assumptions, our algorithms provide an explicit and efficient reduction from hyperelliptic DLPs to nonhyperelliptic DLPs for around 18.57 % of all hyperelliptic genus 3 curves over a given finite field. 1
Exhibiting Sha[2] on hyperelliptic Jacobians
 J. Number Theory
, 2006
"... Abstract. We discuss approaches to computing in the ShafarevichTate group of Jacobians of higher genus curves, with an emphasis on the theory and practice of visualisation. Especially for hyperelliptic curves, this often enables the computation of ranks of Jacobians, even when the 2Selmer bound do ..."
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Cited by 10 (4 self)
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Abstract. We discuss approaches to computing in the ShafarevichTate group of Jacobians of higher genus curves, with an emphasis on the theory and practice of visualisation. Especially for hyperelliptic curves, this often enables the computation of ranks of Jacobians, even when the 2Selmer bound does not bound the rank sharply. This was previously only possible for a few special cases. For curves of genus 2, we also demonstrate a connection with degree 4 del Pezzo surfaces, and show how the BrauerManin obstruction on these surfaces can be used to compute members of the ShafarevichTate group of Jacobians. We derive an explicit parametrised infinite family of genus 2 curves whose Jacobians have nontrivial members of the SharevichTate group. Finally we prove that under certain conditions, the visualisation dimension for order 2 cocycles of Jacobians of certain genus 2 curves is 4 rather than the general bound of 32. 1.
The padic cmmethod for genus 2
, 2005
"... Abstract. We present a nonarchimedian method to construct hyperelliptic CMcurves of genus 2 over finite prime fields. Throughout the document we use the following conventions (this is only for the reference and use of the authors): d degree of the base field of the curve, i.e. C/F2d s number of iso ..."
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Abstract. We present a nonarchimedian method to construct hyperelliptic CMcurves of genus 2 over finite prime fields. Throughout the document we use the following conventions (this is only for the reference and use of the authors): d degree of the base field of the curve, i.e. C/F2d s number of isomorphism classes, in elliptic curve case s = hK n degree of an irreducible component of class invariants K a CM field K0 the real subfield of K K ∗ the reflex CM field of K K ∗ 0 the real subfield of K ∗ j1 absolute Igusa invariant J5 −1
Formulas for the arithmetic geometric mean of curves of genus 3: arXiV:math.AG/0403182
"... Abstract. The arithmetic geometric mean algorithm for calculation of elliptic integrals of the first type was introduced by Gauss. The analog algorithm for Abelian integrals of genus 2 was introduced by Richelot (1837) and Humbert (1901). We present the analogous algorithm for Abelian integrals of g ..."
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Cited by 3 (2 self)
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Abstract. The arithmetic geometric mean algorithm for calculation of elliptic integrals of the first type was introduced by Gauss. The analog algorithm for Abelian integrals of genus 2 was introduced by Richelot (1837) and Humbert (1901). We present the analogous algorithm for Abelian integrals of genus 3. 1.
MODULAR POLYNOMIALS FOR GENUS 2
, 2009
"... Modular polynomials are an important tool in many algorithms involving elliptic curves. In this article we investigate their generalization to the genus 2 case following pioneering work by Gaudry and Dupont. We prove various properties of these genus 2 modular polynomials and give an improved way to ..."
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Modular polynomials are an important tool in many algorithms involving elliptic curves. In this article we investigate their generalization to the genus 2 case following pioneering work by Gaudry and Dupont. We prove various properties of these genus 2 modular polynomials and give an improved way to explicitly compute them.