From Surf Wiki (app.surf) — the open knowledge base

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:

: 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 p3 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. ## 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. ::callout[type=info title="Wikipedia Source"] This article was imported from [Wikipedia](https://en.wikipedia.org/wiki/Division_polynomials) and is available under the [Creative Commons Attribution-ShareAlike 4.0 License](https://creativecommons.org/licenses/by-sa/4.0/). Content has been adapted to SurfDoc format. Original contributors can be found on the [article history page](https://en.wikipedia.org/wiki/Division_polynomials?action=history). ::

polynomials algebraic-curves

Want to explore this topic further?

Ask Mako anything about Division polynomials — get instant answers, deeper analysis, and related topics.

Research with Mako

Free with your Surf account

Content sourced from Wikipedia, available under CC BY-SA 4.0.

This content may have been generated or modified by AI. CloudSurf Software LLC is not responsible for the accuracy, completeness, or reliability of AI-generated content. Always verify important information from primary sources.

Report