793 research outputs found
Fast genus 2 arithmetic based on Theta functions
descriptionInternational audienceIn 1986, D. V. Chudnovsky and G. V. Chudnovsky proposed to use formulae coming from Theta functions for the arithmetic in Jacobians of genus 2 curves. We follow this idea and derive fast formulae for the scalar multiplication in the Kummer surface associated to a genus 2 curve, using a Montgomery ladder. Our formulae can be used to design very efficient genus 2 cryptosystems that should be faster than elliptic curve cryptosystems in some hardware configurations
Fast, uniform, and compact scalar multiplication for elliptic curves and genus 2 Jacobians with applications to signature schemes
We give a general framework for uniform, constant-time one-and
two-dimensional scalar multiplication algorithms for elliptic curves and
Jacobians of genus 2 curves that operate by projecting to the x-line or Kummer
surface, where we can exploit faster and more uniform pseudomultiplication,
before recovering the proper "signed" output back on the curve or Jacobian.
This extends the work of L{\'o}pez and Dahab, Okeya and Sakurai, and Brier and
Joye to genus 2, and also to two-dimensional scalar multiplication. Our results
show that many existing fast pseudomultiplication implementations (hitherto
limited to applications in Diffie--Hellman key exchange) can be wrapped with
simple and efficient pre-and post-computations to yield competitive full scalar
multiplication algorithms, ready for use in more general discrete
logarithm-based cryptosystems, including signature schemes. This is especially
interesting for genus 2, where Kummer surfaces can outperform comparable
elliptic curve systems. As an example, we construct an instance of the Schnorr
signature scheme driven by Kummer surface arithmetic
Effectivity of Arakelov divisors and the theta divisor of a number field
We introduce the notion of an effective Arakelov divisor for a number field
and the arithmetical analogue of the dimension of the space of sections of a
line bundle. We study the analogue of the theta divisor for a number field.Comment: Plain Tex with 5 figures, 21 pages, revised versio
Modular embeddings of Teichmueller curves
Fuchsian groups with a modular embedding have the richest arithmetic
properties among non-arithmetic Fuchsian groups. But they are very rare, all
known examples being related either to triangle groups or to Teichmueller
curves.
In Part I of this paper we study the arithmetic properties of the modular
embedding and develop from scratch a theory of twisted modular forms for
Fuchsian groups with a modular embedding, proving dimension formulas,
coefficient growth estimates and differential equations.
In Part II we provide a modular proof for an Apery-like integrality statement
for solutions of Picard-Fuchs equations. We illustrate the theory on a worked
example, giving explicit Fourier expansions of twisted modular forms and the
equation of a Teichmueller curve in a Hilbert modular surface.
In Part III we show that genus two Teichmueller curves are cut out in Hilbert
modular surfaces by a product of theta derivatives. We rederive most of the
known properties of those Teichmueller curves from this viewpoint, without
using the theory of flat surfaces. As a consequence we give the modular
embeddings for all genus two Teichmueller curves and prove that the Fourier
developments of their twisted modular forms are algebraic up to one
transcendental scaling constant. Moreover, we prove that Bainbridge's
compactification of Hilbert modular surfaces is toroidal. The strategy to
compactify can be expressed using continued fractions and resembles
Hirzebruch's in form, but every detail is different.Comment: revision including the referee's comments, to appear in Compositio
Mat
A p-adic quasi-quadratic point counting algorithm
In this article we give an algorithm for the computation of the number of
rational points on the Jacobian variety of a generic ordinary hyperelliptic
curve defined over a finite field of cardinality with time complexity
and space complexity , where . In the latter
complexity estimate the genus and the characteristic are assumed as fixed. Our
algorithm forms a generalization of both, the AGM algorithm of J.-F. Mestre and
the canonical lifting method of T. Satoh. We canonically lift a certain
arithmetic invariant of the Jacobian of the hyperelliptic curve in terms of
theta constants. The theta null values are computed with respect to a
semi-canonical theta structure of level where is an integer
and p=\mathrm{char}(\F_q)>2. The results of this paper suggest a global
positive answer to the question whether there exists a quasi-quadratic time
algorithm for the computation of the number of rational points on a generic
ordinary abelian variety defined over a finite field.Comment: 32 page
Computing isogenies between Jacobian of curves of genus 2 and 3
We present a quasi-linear algorithm to compute isogenies between Jacobians of
curves of genus 2 and 3 starting from the equation of the curve and a maximal
isotropic subgroup of the l-torsion, for l an odd prime number, generalizing
the V\'elu's formula of genus 1. This work is based from the paper "Computing
functions on Jacobians and their quotients" of Jean-Marc Couveignes and Tony
Ezome. We improve their genus 2 case algorithm, generalize it for genus 3
hyperelliptic curves and introduce a way to deal with the genus 3
non-hyperelliptic case, using algebraic theta functions.Comment: 34 page
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
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