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 $$\mathbb{Z}[x,y,A,B]$$ with $$x, y, A, B$$ free variables that is recursively defined by:


 * $$\psi_{0} = 0 $$


 * $$\psi_{1} = 1$$


 * $$\psi_{2} = 2y$$


 * $$\psi_{3} = 3x^{4} + 6Ax^{2} + 12Bx - A^{2}$$


 * $$\psi_{4} = 4y(x^{6} + 5Ax^{4} + 20Bx^{3} - 5A^{2}x^{2} - 4ABx - 8B^{2} - A^{3}) $$


 * $$\vdots$$


 * $$\psi_{2m+1} = \psi_{m+2} \psi_{m}^{ 3}  -  \psi_{m-1} \psi ^{ 3}_{ m+1} \text{ for } m \geq 2$$


 * $$\psi_{ 2m} = \left ( \frac { \psi_{m}}{2y} \right ) \cdot ( \psi_{m+2}\psi^{ 2}_{m-1} -  \psi_{m-2} \psi ^{ 2}_{m+1})   \text{ for } m \geq 3$$

The polynomial $$\psi_n$$ is called the nth division polynomial.

Properties

 * In practice, one sets $$y^2=x^3+Ax+B$$, and then $$\psi_{2m+1}\in\mathbb{Z}[x,A,B]$$ and $$\psi_{2m}\in 2y\mathbb{Z}[x,A,B]$$.
 * The division polynomials form a generic elliptic divisibility sequence over the ring $$\mathbb{Q}[x,y,A,B]/(y^2-x^3-Ax-B)$$.
 * If an elliptic curve $$E$$ is given in the Weierstrass form $$y^2=x^3+Ax+B$$ over some field $$K$$, i.e. $$A, B\in K$$, one can use these values of $$A, B$$ and consider the division polynomials in the coordinate ring of $$E$$. The roots of $$\psi_{2n+1}$$ are the $$x$$-coordinates of the points of $$E[2n+1]\setminus \{O\}$$, where $$E[2n+1]$$ is the $$(2n+1)^{\text{th}}$$ torsion subgroup of $$E$$. Similarly, the roots of $$\psi_{2n}/y$$ are the $$x$$-coordinates of the points of $$E[2n]\setminus E[2]$$.
 * Given a point $$P=(x_P,y_P)$$ on the elliptic curve $$E:y^2=x^3+Ax+B$$ over some field $$K$$, we can express the coordinates of the nth multiple of $$P$$ in terms of division polynomials:
 * $$nP= \left ( \frac{\phi_{n}(x)}{\psi_{n}^{2}(x)}, \frac{\omega_{n}(x,y)}{\psi^{3}_{n}(x,y)} \right) =  \left( x - \frac {\psi_{n-1} \psi_{n+1}}{\psi^{2}_{n}(x)}, \frac{\psi_{2 n}(x,y)}{2\psi^{4}_{n}(x)} \right)$$
 * where $$\phi_{n}$$ and $$\omega_{n}$$ are defined by:
 * $$\phi_{n}=x\psi_{n}^{2} - \psi_{n+1}\psi_{n-1},$$
 * $$\omega_{n}=\frac{\psi_{n+2}\psi_{n-1}^{2}-\psi_{n-2}\psi_{n+1}^{2}}{4y}.$$

Using the relation between $$\psi_{2m}$$ and $$\psi _{2m + 1}$$, along with the equation of the curve, the functions $$\psi_{n}^{2}$$, $$\frac{\psi_{2n}}{y}, \psi_{2n + 1}$$, $$\phi_{n}$$ are all in $$K[x]$$.

Let $$p>3$$ be prime and let $$E:y^2=x^3+Ax+B$$ be an elliptic curve over the finite field $$\mathbb{F}_p$$, i.e., $$A,B \in \mathbb{F}_p$$. The $$\ell$$-torsion group of $$E$$ over $$\bar{ \mathbb{F}}_p$$ is isomorphic to $$\mathbb{Z}/\ell \times \mathbb{Z}/\ell$$ if $$\ell\neq p$$, and to $$\mathbb{Z}/\ell $$ or $$\{0\}$$ if $$\ell=p$$. Hence the degree of $$\psi_\ell$$ is equal to either $$\frac{1}{2}(l^2-1)$$, $$\frac{1}{2}(l-1)$$, or 0.

René Schoof observed that working modulo the $$\ell$$th 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.