Computing the $\ell$-power torsion of an elliptic curve over a finite field

The algorithm we develop outputs the order and the structure, including generators, of the $\ell$-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 $\ell$-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 $\ell$. We specify in complete detail the case $\ell=3$, when the complexity of each trisection is given by the computation of cubic roots in finite fields.