Abstract
We study the problem of efficiently constructing a curve
$C$
of genus
$2$
over a finite field
$\mathbb{F}$
for which either the curve
$C$
itself or its Jacobian has a prescribed number
$N$
of
$\mathbb{F}$
-rational points.
In the case of the Jacobian, we show that any ‘CM-construction’ to produce the required genus-
$2$
curves necessarily takes time exponential in the size of its input.
On the other hand, we provide an algorithm for producing a genus-
$2$
curve with a given number of points that, heuristically, takes polynomial time for most input values. We illustrate the practical applicability of this algorithm by constructing a genus-
$2$
curve having exactly
$10^{2014}+9703$
(prime) points, and two genus-
$2$
curves each having exactly
$10^{2013}$
points.
In an appendix we provide a complete parametrization, over an arbitrary base field
$k$
of characteristic neither two nor three, of the family of genus-
$2$
curves over
$k$
that have
$k$
-rational degree-
$3$
maps to elliptic curves, including formulas for the genus-
$2$
curves, the associated elliptic curves, and the degree-
$3$
maps.
Supplementary materials are available with this article.