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 with x, y, A, B free variables that is recursively defined by: The polynomial \psi_n is called the nth division polynomial.

Properties

Using the relation between \psi_{2m} and, along with the equation of the curve, the functions , , \phi_{n} are all in K[x]. Let p>3 be prime and let be an elliptic curve over the finite field, i.e.,. The \ell-torsion group of E over is isomorphic to if \ell\neq p, and to or {0} if \ell=p. Hence the degree of \psi_\ell is equal to either, , or 0. René Schoof observed that working modulo the \ellth division polynomial allows one to work with all \ell-torsion points simultaneously. This is heavily used in Schoof's algorithm for counting points on elliptic curves.