We study the problem of efficiently constructing a curve
over a finite field
for which either the curve
itself or its Jacobian has a prescribed number
In the case of the Jacobian, we show that any ‘CM-construction’ to produce the required genus-
curves necessarily takes time exponential in the size of its input.
On the other hand, we provide an algorithm for producing a genus-
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-
curve having exactly
(prime) points, and two genus-
curves each having exactly
In an appendix we provide a complete parametrization, over an arbitrary base field
of characteristic neither two nor three, of the family of genus-
maps to elliptic curves, including formulas for the genus-
curves, the associated elliptic curves, and the degree-
Supplementary materials are available with this article.