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.


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.


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.

See also


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