Generators for the l-Torsion Subgroup of Jacobians of Genus Two Curves

We give an explicit description of the matrix representation of the Frobenius endomorphism on the Jacobian of a genus two curve on the subgroup of l-torsion points. By using this description, we can describe the matrix representation of the Weil-pairing on the subgroup of e-torsion points explicitly. Finally, the explicit description of the Weil-pairing provides us with an efficient, probabilistic algorithm to find generators of the subgroup of l-torsion points on the Jacobian of a genus two curve.