Computing the ℓ-power torsion of an elliptic curve over a finite field
Josep M. Miret, R. Peña-Moreno, Anna Rio, Magda Valls · Mathematics of Computation · 2008
The algorithm we develop outputs the order and the structure, including generators, of the ℓ \ell -Sylow subgroup of the group of rational points of an elliptic curve defined over a finite field. To do this, we do not assume any knowledge of the group order. We are able to choose points in such a way that a linear number of successive ℓ \ell -divisions leads to generators of the subgroup under consideration. After the computation of a couple of polynomials, each division step relies on finding rational roots of polynomials of degree ℓ \ell . We specify in complete detail the case ℓ = 3 \ell =3 , when the complexity of each trisection is given by the computation of cubic roots in finite fields.