6,154 research outputs found
Effective reconstruction of generic genus 4 curves from their theta hyperplanes
Effective reconstruction formulas of a curve from its theta hyperplanes are
known classically in genus 2 (where the theta hyperplanes are Weierstrass
points), and 3 (where, for a generic curve, the theta hyperplanes are
bitangents to a plane quartic). However, for higher genera, no formula or
algorithm are known. In this paper we give an explicit (and simple) algorithm
for computing a generic genus 4 curve from it's theta hyperplanes.Comment: no content modification to previous version; presentation
modification following referees comment
Isogenies of Elliptic Curves: A Computational Approach
Isogenies, the mappings of elliptic curves, have become a useful tool in
cryptology. These mathematical objects have been proposed for use in computing
pairings, constructing hash functions and random number generators, and
analyzing the reducibility of the elliptic curve discrete logarithm problem.
With such diverse uses, understanding these objects is important for anyone
interested in the field of elliptic curve cryptography. This paper, targeted at
an audience with a knowledge of the basic theory of elliptic curves, provides
an introduction to the necessary theoretical background for understanding what
isogenies are and their basic properties. This theoretical background is used
to explain some of the basic computational tasks associated with isogenies.
Herein, algorithms for computing isogenies are collected and presented with
proofs of correctness and complexity analyses. As opposed to the complex
analytic approach provided in most texts on the subject, the proofs in this
paper are primarily algebraic in nature. This provides alternate explanations
that some with a more concrete or computational bias may find more clear.Comment: Submitted as a Masters Thesis in the Mathematics department of the
University of Washingto
Computing local p-adic height pairings on hyperelliptic curves
We describe an algorithm to compute the local component at p of the
Coleman-Gross p-adic height pairing on divisors on hyperelliptic curves. As the
height pairing is given in terms of a Coleman integral, we also provide new
techniques to evaluate Coleman integrals of meromorphic differentials and
present our algorithms as implemented in Sage
Computing low-degree isogenies in genus 2 with the Dolgachev-Lehavi method
Let ell be a prime, and H a curve of genus 2 over a field k of characteristic
not 2 or ell. If S is a maximal Weil-isotropic subgroup of Jac(H)[ell], then
Jac(H)/S is isomorphic to the Jacobian of some (possibly reducible) curve X. We
investigate the Dolgachev--Lehavi method for constructing the curve X,
simplifying their approach and making it more explicit. The result, at least
for ell=3, is an efficient and easily programmable algorithm suitable for
number-theoretic calculations
Fast algorithms for computing isogenies between elliptic curves
We survey algorithms for computing isogenies between elliptic curves defined
over a field of characteristic either 0 or a large prime. We introduce a new
algorithm that computes an isogeny of degree ( different from the
characteristic) in time quasi-linear with respect to . This is based in
particular on fast algorithms for power series expansion of the Weierstrass
-function and related functions
Faster computation of the Tate pairing
This paper proposes new explicit formulas for the doubling and addition step
in Miller's algorithm to compute the Tate pairing. For Edwards curves the
formulas come from a new way of seeing the arithmetic. We state the first
geometric interpretation of the group law on Edwards curves by presenting the
functions which arise in the addition and doubling. Computing the coefficients
of the functions and the sum or double of the points is faster than with all
previously proposed formulas for pairings on Edwards curves. They are even
competitive with all published formulas for pairing computation on Weierstrass
curves. We also speed up pairing computation on Weierstrass curves in Jacobian
coordinates. Finally, we present several examples of pairing-friendly Edwards
curves.Comment: 15 pages, 2 figures. Final version accepted for publication in
Journal of Number Theor
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