892 research outputs found
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
On the closure of elliptic wedge operators
We prove a semi-Fredholm theorem for the minimal extension of elliptic
operators on manifolds with wedge singularities and give, under suitable
assumptions, a full asymptotic expansion of the trace of the resolvent.Comment: 22 pages, improved expositio
Index calculus for abelian varieties of small dimension and the elliptic curve discrete logarithm problem
International audienceWe propose an index calculus algorithm for the discrete logarithm problem on general abelian varieties of small dimension. The main difference with the previous approaches is that we do not make use of any embedding into the Jacobian of a well-suited curve. We apply this algorithm to the Weil restriction of elliptic curves and hyperelliptic curves over small degree extension fields. In particular, our attack can solve an elliptic curve discrete logarithm problem defined over GF(q^3) in heuristic asymptotic running time O~(q^(4/3)); and an elliptic problem over GF(q^4) or a genus 2 problem over GF(q^2) in heuristic asymptotic running time O~(q^(3/2))
- …