# 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 with free variables that is recursively defined by:

The polynomial is called the nth division polynomial.

## Properties

• In practice, one sets , and then and .
• The division polynomials form a generic elliptic divisibility sequence over the ring .
• If an elliptic curve is given in the Weierstrass form over some field , i.e. , one can use these values of and consider the division polynomials in the coordinate ring of . The roots of are the -coordinates of the points of , where is the torsion subgroup of . Similarly, the roots of are the -coordinates of the points of .
• Given a point on the elliptic curve over some field , we can express the coordinates of the nth multiple of in terms of division polynomials:
where and are defined by:

Using the relation between and , along with the equation of the curve, the functions , and are all in .

Let be prime and let be an elliptic curve over the finite field , i.e., . The -torsion group of over is isomorphic to if , and to or if . Hence the degree of is equal to either , , or 0.

René Schoof observed that working modulo the th division polynomial allows one to work with all -torsion points simultaneously. This is heavily used in Schoof's algorithm for counting points on elliptic curves.