Summary: It builds an algorithm that determines the l-subgrupos of Sylow order l cousin, Group E (R) points of a elliptic curve defined by a body finite F. The algorithm accepts as input an elliptic curve and cousin l. And returns one or two points of the sub-Sylow generators and their respective orders. The algorithm is constructed by combining the sub-Sylow few trees rooted at the point of infinity and whose nodes are the points of l-subgrupo of Sylow. The edges are defined by pairs of points (Q, P), such that [t] P = Q. Every step of the algorithm consists of a âdescensoâ by the edge (Q, P), such that known point Q is determining Q: We call that determination l-división Q. The algorithm begins with the points of the sub-l-torsión of the curve and ends when they reach their maximum height of the tree. For cases l = 2, 3, each fall by a ridge has been resolved by calculating character and quadratic and cubic roots. In the general case, ie when l> 3, these steps represent the calculation in F of the roots of two polynomials of degree l. The identification and study of such polynomials effective has been done about widespread expressions of Vélu (
1971) for abscisa point isógeno P, the isogenia the core of which is generated by a group cyclic rational point of order l, which from the start the algorithm, we know that it exists. It also has identified the types of factoring polynomial l-división of elliptic curves defined over finite bodies, when you have a rational point of order l. Likewise, other types of factoring polynomial associated with l-división, grade Square l, which we called l-isogenia. We have studied the costs of the different algorithms, which are polinómicos being in the order of the body of the definition of Elliptic Curve.
