762 research outputs found
Pointless curves of genus three and four
A curve over a field k is pointless if it has no k-rational points. We show
that there exist pointless genus-3 hyperelliptic curves over a finite field F_q
if and only if q < 26, that there exist pointless smooth plane quartics over
F_q if and only if either q < 24 or q = 29 or q = 32, and that there exist
pointless genus-4 curves over F_q if and only if q < 50.Comment: LaTeX, 15 page
Counting hyperelliptic curves that admit a Koblitz model
Let k be a finite field of odd characteristic. We find a closed formula for
the number of k-isomorphism classes of pointed, and non-pointed, hyperelliptic
curves of genus g over k, admitting a Koblitz model. These numbers are
expressed as a polynomial in the cardinality q of k, with integer coefficients
(for pointed curves) and rational coefficients (for non-pointed curves). The
coefficients depend on g and the set of divisors of q-1 and q+1. These formulas
show that the number of hyperelliptic curves of genus g suitable (in principle)
of cryptographic applications is asymptotically (1-e^{-1})2q^{2g-1}, and not
2q^{2g-1} as it was believed. The curves of genus g=2 and g=3 are more
resistant to the attacks to the DLP; for these values of g the number of curves
is respectively (91/72)q^3+O(q^2) and (3641/2880)q^5+O(q^4)
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
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