35 research outputs found
Liftings of Reduction Maps for Quaternion Algebras
We construct liftings of reduction maps from CM points to supersingular
points for general quaternion algebras and use these liftings to establish a
precise correspondence between CM points on indefinite quaternion algebras with
a given conductor and CM points on certain corresponding totally definite
quaternion algebras.Comment: 17 page
Horizontal isogeny graphs of ordinary abelian varieties and the discrete logarithm problem
Fix an ordinary abelian variety defined over a finite field. The ideal class
group of its endomorphism ring acts freely on the set of isogenous varieties
with same endomorphism ring, by complex multiplication. Any subgroup of the
class group, and generating set thereof, induces an isogeny graph on the orbit
of the variety for this subgroup. We compute (under the Generalized Riemann
Hypothesis) some bounds on the norms of prime ideals generating it, such that
the associated graph has good expansion properties.
We use these graphs, together with a recent algorithm of Dudeanu, Jetchev and
Robert for computing explicit isogenies in genus 2, to prove random
self-reducibility of the discrete logarithm problem within the subclasses of
principally polarizable ordinary abelian surfaces with fixed endomorphism ring.
In addition, we remove the heuristics in the complexity analysis of an
algorithm of Galbraith for explicitly computing isogenies between two elliptic
curves in the same isogeny class, and extend it to a more general setting
including genus 2.Comment: 18 page
Global Divisibility of Heegner Points and Tamagawa Numbers
We improve Kolyvagin's upper bound on the order of the -primary part of
the Shafarevich-Tate group of an elliptic curve of rank one over a quadratic
imaginary field. In many cases, our bound is precisely the one predicted by the
Birch and Swinnerton-Dyer conjectural formula.Comment: 20 page
Isogeny graphs of ordinary abelian varieties
Fix a prime number . Graphs of isogenies of degree a power of
are well-understood for elliptic curves, but not for higher-dimensional abelian
varieties. We study the case of absolutely simple ordinary abelian varieties
over a finite field. We analyse graphs of so-called -isogenies,
resolving that they are (almost) volcanoes in any dimension. Specializing to
the case of principally polarizable abelian surfaces, we then exploit this
structure to describe graphs of a particular class of isogenies known as
-isogenies: those whose kernels are maximal isotropic subgroups
of the -torsion for the Weil pairing. We use these two results to write
an algorithm giving a path of computable isogenies from an arbitrary absolutely
simple ordinary abelian surface towards one with maximal endomorphism ring,
which has immediate consequences for the CM-method in genus 2, for computing
explicit isogenies, and for the random self-reducibility of the discrete
logarithm problem in genus 2 cryptography.Comment: 36 pages, 4 figure
Collision bounds for the additive Pollard rho algorithm for solving discrete logarithms
We prove collision bounds for the Pollard rho algorithm to solve the discrete logarithm problem in a general cyclic group . Unlike the setting studied by Kim et al., we consider additive walks: the setting used in practice to solve the elliptic curve discrete logarithm problem. Our bounds differ from the birthday bound (||) by a factor of log|| and are based on mixing time estimates for random walks on finite abelian groups due to Dou and Hildebran