Division polynomials

In mathematics the division polynomials provide a way to calculate multiples of points on elliptic curves and to study the fields generated by torsion points. They play a central role in the study of counting points on elliptic curves in Schoof's algorithm.
Definition.
The set of division polynomials is a sequence of polynomials in formula_1 with formula_2 free variables that is recursively defined by:
The polynomial formula_11 is called the "n"th division polynomial.
Properties.
Using the relation between formula_40 and formula_41, along with the equation of the curve, the functions formula_42 , formula_43, formula_36 are all in formula_45.
Let formula_46 be prime and let formula_32 be an elliptic curve over the finite field formula_48, i.e., formula_49. The formula_50-torsion group of formula_16 over formula_52 is isomorphic to formula_53 if formula_54, and to formula_55 or formula_56 if formula_57. Hence the degree of formula_58 is equal to either formula_59, formula_60, or 0.
René Schoof observed that working modulo the formula_50"th" division polynomial allows one to work with all formula_50-torsion points simultaneously. This is heavily used in Schoof's algorithm for counting points on elliptic curves.