346 research outputs found
The geometry of efficient arithmetic on elliptic curves
The arithmetic of elliptic curves, namely polynomial addition and scalar
multiplication, can be described in terms of global sections of line bundles on
and , respectively, with respect to a given projective embedding
of in . By means of a study of the finite dimensional vector
spaces of global sections, we reduce the problem of constructing and finding
efficiently computable polynomial maps defining the addition morphism or
isogenies to linear algebra. We demonstrate the effectiveness of the method by
improving the best known complexity for doubling and tripling, by considering
families of elliptic curves admiting a -torsion or -torsion point
Addition law structure of elliptic curves
The study of alternative models for elliptic curves has found recent interest
from cryptographic applications, once it was recognized that such models
provide more efficiently computable algorithms for the group law than the
standard Weierstrass model. Examples of such models arise via symmetries
induced by a rational torsion structure. We analyze the module structure of the
space of sections of the addition morphisms, determine explicit dimension
formulas for the spaces of sections and their eigenspaces under the action of
torsion groups, and apply this to specific models of elliptic curves with
parametrized torsion subgroups
Higher dimensional 3-adic CM construction
We find equations for the higher dimensional analogue of the modular curve
X_0(3) using Mumford's algebraic formalism of algebraic theta functions. As a
consequence, we derive a method for the construction of genus 2 hyperelliptic
curves over small degree number fields whose Jacobian has complex
multiplication and good ordinary reduction at the prime 3. We prove the
existence of a quasi-quadratic time algorithm for computing a canonical lift in
characteristic 3 based on these equations, with a detailed description of our
method in genus 1 and 2.Comment: 23 pages; major revie
The Weierstrass subgroup of a curve has maximal rank
We show that the Weierstrass points of the generic curve of genus over an
algebraically closed field of characteristic 0 generate a group of maximal rank
in the Jacobian
Complete addition laws on abelian varieties
We prove that under any projective embedding of an abelian variety A of
dimension g, a complete system of addition laws has cardinality at least g+1,
generalizing of a result of Bosma and Lenstra for the Weierstrass model of an
elliptic curve in P^2. In contrast with this geometric constraint, we moreover
prove that if k is any field with infinite absolute Galois group, then there
exists, for every abelian variety A/k, a projective embedding and an addition
law defined for every pair of k-rational points. For an abelian variety of
dimension 1 or 2, we show that this embedding can be the classical Weierstrass
model or embedding in P^15, respectively, up to a finite number of
counterexamples for |k| less or equal to 5.Comment: 9 pages. Finale version, accepted for publication in LMS Journal of
Computation and Mathematic
Arithmetic statistics of Galois groups
We develop a computational framework for the statistical characterization of
Galois characters with finite image, with application to characterizing Galois
groups and establishing equivalence of characters of finite images of
On the quaternion -isogeny path problem
Let \cO be a maximal order in a definite quaternion algebra over
of prime discriminant , and a small prime. We describe a
probabilistic algorithm, which for a given left -ideal, computes a
representative in its left ideal class of -power norm. In practice the
algorithm is efficient, and subject to heuristics on expected distributions of
primes, runs in expected polynomial time. This breaks the underlying problem
for a quaternion analog of the Charles-Goren-Lauter hash function, and has
security implications for the original CGL construction in terms of
supersingular elliptic curves.Comment: To appear in the LMS Journal of Computation and Mathematics, as a
special issue for ANTS (Algorithmic Number Theory Symposium) conferenc
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