%0 Journal Article
%T Genus-2 curves and Jacobians with a given number of points
%A Reinier Br£¿ker
%A Everett W. Howe
%A Kristin E. Lauter
%A Peter Stevenhagen
%J Mathematics
%D 2014
%I arXiv
%R 10.1112/S1461157014000461
%X We study the problem of efficiently constructing a curve C of genus 2 over a finite field F for which either the curve C itself or its Jacobian has a prescribed number N of 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 2 nor 3, 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.
%U http://arxiv.org/abs/1403.6911v3