405 research outputs found
Fast algorithms for computing isogenies between ordinary elliptic curves in small characteristic
The problem of computing an explicit isogeny between two given elliptic
curves over F_q, originally motivated by point counting, has recently awaken
new interest in the cryptology community thanks to the works of Teske and
Rostovstev & Stolbunov.
While the large characteristic case is well understood, only suboptimal
algorithms are known in small characteristic; they are due to Couveignes,
Lercier, Lercier & Joux and Lercier & Sirvent. In this paper we discuss the
differences between them and run some comparative experiments. We also present
the first complete implementation of Couveignes' second algorithm and present
improvements that make it the algorithm having the best asymptotic complexity
in the degree of the isogeny.Comment: 21 pages, 6 figures, 1 table. Submitted to J. Number Theor
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
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 cardinalities of Q-curve reductions over finite fields
We present a specialized point-counting algorithm for a class of elliptic
curves over F\_{p^2} that includes reductions of quadratic Q-curves modulo
inert primes and, more generally, any elliptic curve over F\_{p^2} with a
low-degree isogeny to its Galois conjugate curve. These curves have interesting
cryptographic applications. Our algorithm is a variant of the
Schoof--Elkies--Atkin (SEA) algorithm, but with a new, lower-degree
endomorphism in place of Frobenius. While it has the same asymptotic asymptotic
complexity as SEA, our algorithm is much faster in practice.Comment: To appear in the proceedings of ANTS-XII. Added acknowledgement of
Drew Sutherlan
Constructing Permutation Rational Functions From Isogenies
A permutation rational function is a rational function
that induces a bijection on , that is, for all
there exists exactly one such that . Permutation
rational functions are intimately related to exceptional rational functions,
and more generally exceptional covers of the projective line, of which they
form the first important example.
In this paper, we show how to efficiently generate many permutation rational
functions over large finite fields using isogenies of elliptic curves, and
discuss some cryptographic applications. Our algorithm is based on Fried's
modular interpretation of certain dihedral exceptional covers of the projective
line (Cont. Math., 1994)
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
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
Fast Arithmetics in Artin-Schreier Towers over Finite Fields
An Artin-Schreier tower over the finite field F_p is a tower of field
extensions generated by polynomials of the form X^p - X - a. Following Cantor
and Couveignes, we give algorithms with quasi-linear time complexity for
arithmetic operations in such towers. As an application, we present an
implementation of Couveignes' algorithm for computing isogenies between
elliptic curves using the p-torsion.Comment: 28 pages, 4 figures, 3 tables, uses mathdots.sty, yjsco.sty Submitted
to J. Symb. Compu
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