Paper 2008/491

A CM construction for curves of genus 2 with p-rank 1

Laura Hitt O'Connor, Gary McGuire, Michael Naehrig, and Marco Streng

Abstract

We construct Weil numbers corresponding to genus-2 curves with $p$-rank 1 over the finite field ${\text{F}}_{p^2}$ of $p^2$ elements. The corresponding curves can be constructed using explicit CM constructions. In one of our algorithms, the group of ${\text{F}}_{p^2}$-valued points of the Jacobian has prime order, while another allows for a prescribed embedding degree with respect to a subgroup of prescribed order. The curves are defined over ${\text{F}}_{p^2}$ out of necessity: we show that curves of $p$-rank 1 over ${\text{F}}_p$ for large $p$ cannot be efficiently constructed using explicit CM constructions.