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:

$\psi _{0}=0$

$\psi _{1}=1$

$\psi _{2}=2y$

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

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

$\vdots$

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

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

The polynomial $\psi _{n}$ is called the n^th 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 n^th multiple of $P$ in terms of division polynomials:

$nP=\left({\frac {\phi _{n}(x)}{\psi _{n}^{2}(x)}},{\frac {\omega _{n}(x,y)}{\psi _{n}^{3}(x,y)}}\right)=\left(x-{\frac {\psi _{n-1}\psi _{n+1}}{\psi _{n}^{2}(x)}},{\frac {\psi _{2n}(x,y)}{2\psi _{n}^{4}(x)}}\right)$

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.

See also

• Schoof's algorithm

References

• A. Enge: Elliptic Curves and their Applications to Cryptography: An Introduction. Kluwer Academic Publishers, Dordrecht, 1999.
• N. Koblitz: A Course in Number Theory and Cryptography, Graduate Texts in Math. No. 114, Springer-Verlag, 1987. Second edition, 1994
• Müller : Die Berechnung der Punktanzahl von elliptischen kurven über endlichen Primkörpern. Master's Thesis. Universität des Saarlandes, Saarbrücken, 1991.
• G. Musiker: Schoof's Algorithm for Counting Points on $E(\mathbb {F} _{q})$. Available at https://www-users.cse.umn.edu/~musiker/schoof.pdf
• Schoof: Elliptic Curves over Finite Fields and the Computation of Square Roots mod p. Math. Comp., 44(170):483–494, 1985. Available at http://www.mat.uniroma2.it/~schoof/ctpts.pdf
• R. Schoof: Counting Points on Elliptic Curves over Finite Fields. J. Theor. Nombres Bordeaux 7:219–254, 1995. Available at http://www.mat.uniroma2.it/~schoof/ctg.pdf
• L. C. Washington: Elliptic Curves: Number Theory and Cryptography. Chapman & Hall/CRC, New York, 2003.
• J. Silverman: The Arithmetic of Elliptic Curves, Springer-Verlag, GTM 106, 1986.