Generators for the ℓ-torsion subgroup of Jacobians of genus two curves

Abstract

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 ℓ-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. © Springer-Verlag Berlin Heidelberg 2008.