We construct Weil numbers corresponding to genus-2 curves with p-rank 1
over the finite field \F_{p^2} of p2 elements. The corresponding curves
can be constructed using explicit CM constructions. In one of our algorithms,
the group of \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 \F_{p^2} out of necessity: we
show that curves of p-rank 1 over \F_p for large p cannot be efficiently
constructed using explicit CM constructions.Comment: 19 page