93 research outputs found
Isogenies and the Discrete Logarithm Problem in Jacobians of Genus 3 Hyperelliptic Curves
We describe the use of explicit isogenies to translate instances of the
Discrete Logarithm Problem (DLP) from Jacobians of hyperelliptic genus 3 curves
to Jacobians of non-hyperelliptic genus 3 curves, where they are vulnerable to
faster index calculus attacks. We provide explicit formulae for isogenies with
kernel isomorphic to (\ZZ/2\ZZ)^3 (over an algebraic closure of the base
field) for any hyperelliptic genus 3 curve over a field of characteristic not 2
or 3. These isogenies are rational for a positive fraction of all hyperelliptic
genus 3 curves defined over a finite field of characteristic . Subject
to reasonable assumptions, our constructions give an explicit and efficient
reduction of instances of the DLP from hyperelliptic to non-hyperelliptic
Jacobians for around 18.57% of all hyperelliptic genus 3 curves over a given
finite field. We conclude with a discussion on extending these ideas to
isogenies with more general kernels. A condensed version of this work appeared
in the proceedings of the EUROCRYPT 2008 conference.Comment: This is an extended version of work that appeared in the proceedings
of the Eurocrypt 2008 conferenc
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
Curves, Jacobians, and Cryptography
The main purpose of this paper is to give an overview over the theory of
abelian varieties, with main focus on Jacobian varieties of curves reaching
from well-known results till to latest developments and their usage in
cryptography. In the first part we provide the necessary mathematical
background on abelian varieties, their torsion points, Honda-Tate theory,
Galois representations, with emphasis on Jacobian varieties and hyperelliptic
Jacobians. In the second part we focus on applications of abelian varieties on
cryptography and treating separately, elliptic curve cryptography, genus 2 and
3 cryptography, including Diffie-Hellman Key Exchange, index calculus in Picard
groups, isogenies of Jacobians via correspondences and applications to discrete
logarithms. Several open problems and new directions are suggested.Comment: 66 page
Discrete logarithms in curves over finite fields
A survey on algorithms for computing discrete logarithms in Jacobians of
curves over finite fields
Families of explicitly isogenous Jacobians of variable-separated curves
We construct six infinite series of families of pairs of curves (X,Y) of
arbitrarily high genus, defined over number fields, together with an explicit
isogeny from the Jacobian of X to the Jacobian of Y splitting multiplication by
2, 3, or 4. For each family, we compute the isomorphism type of the isogeny
kernel and the dimension of the image of the family in the appropriate moduli
space. The families are derived from Cassou--Nogu\`es and Couveignes' explicit
classification of pairs (f,g) of polynomials such that f(x_1) - g(x_2) is
reducible
A Generic Approach to Searching for Jacobians
We consider the problem of finding cryptographically suitable Jacobians. By
applying a probabilistic generic algorithm to compute the zeta functions of low
genus curves drawn from an arbitrary family, we can search for Jacobians
containing a large subgroup of prime order. For a suitable distribution of
curves, the complexity is subexponential in genus 2, and O(N^{1/12}) in genus
3. We give examples of genus 2 and genus 3 hyperelliptic curves over prime
fields with group orders over 180 bits in size, improving previous results. Our
approach is particularly effective over low-degree extension fields, where in
genus 2 we find Jacobians over F_{p^2) and trace zero varieties over F_{p^3}
with near-prime orders up to 372 bits in size. For p = 2^{61}-1, the average
time to find a group with 244-bit near-prime order is under an hour on a PC.Comment: 22 pages, to appear in Mathematics of Computatio
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
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