257 research outputs found
Constructing genus 3 hyperelliptic Jacobians with CM
Given a sextic CM field , we give an explicit method for finding all genus
3 hyperelliptic curves defined over whose Jacobians are simple and
have complex multiplication by the maximal order of this field, via an
approximation of their Rosenhain invariants. Building on the work of Weng, we
give an algorithm which works in complete generality, for any CM sextic field
, and computes minimal polynomials of the Rosenhain invariants for any
period matrix of the Jacobian. This algorithm can be used to generate genus 3
hyperelliptic curves over a finite field with a given zeta
function by finding roots of the Rosenhain minimal polynomials modulo .Comment: 20 pages; to appear in ANTS XI
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
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
Pairing-based algorithms for jacobians of genus 2 curves with maximal endomorphism ring
Using Galois cohomology, Schmoyer characterizes cryptographic non-trivial
self-pairings of the -Tate pairing in terms of the action of the
Frobenius on the -torsion of the Jacobian of a genus 2 curve. We apply
similar techniques to study the non-degeneracy of the -Tate pairing
restrained to subgroups of the -torsion which are maximal isotropic with
respect to the Weil pairing. First, we deduce a criterion to verify whether the
jacobian of a genus 2 curve has maximal endomorphism ring. Secondly, we derive
a method to construct horizontal -isogenies starting from a
jacobian with maximal endomorphism ring
A database of genus 2 curves over the rational numbers
We describe the construction of a database of genus 2 curves of small
discriminant that includes geometric and arithmetic invariants of each curve,
its Jacobian, and the associated L-function. This data has been incorporated
into the L-Functions and Modular Forms Database (LMFDB).Comment: 15 pages, 7 tables; bibliography formatting and typos fixe
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