Generators of Jacobians of genus two curves

Abstract

We prove that in most cases relevant to cryptography, the Frobenius endomorphism on the Jacobian of a genus two curve is represented by a diagonal matrix with respect to an appropriate basis of the subgroup of l-torsion points. From this fact we get an explicit description of the Weil-pairing on the subgroup of l-torsion points. 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.