﻿

# Division polynomials

In mathematics the division polynomials provide a way to calculate multiples of points on elliptic curves over Finite fields. They play a central role in the study of counting points on elliptic curves in Schoof's algorithm.

## Definition

The division polynomials are a sequence of polynomials in $\mathbb{Z}[x,y,A,B]$ with x,y,A,B free variables that is recursively defined by:

ψ0 = 0
ψ1 = 1
ψ2 = 2y
ψ3 = 3x4 + 6Ax2 + 12BxA2
ψ4 = 4y(x6 + 5Ax4 + 20Bx3 − 5A2x2 − 4ABx − 8B2A3) $\vdots$ $\psi_{2m+1} = \psi_{m+2} \psi_{m}^{ 3} - \psi_{m-1} \psi ^{ 3}_{ m+1} \text{ for } m \geq 2$ $\psi_{ 2m} = \left ( \frac { \psi_{m}}{2y} \right ) \cdot ( \psi_{m+2}\psi^{ 2}_{m-1} - \psi_{m-2} \psi ^{ 2}_{m+1}) \text{ for } m \geq 3$

The polynomial ψn is called the nth division polynomial.

## Properties

• ψ2n + 1 is a polynomial in Z[x,y2,A,B], while ψ2m is a polynomial in 2yZ[x,y2,A,B].
• The division polynomials form an elliptic divisibility sequence. Moreover all nonsingular elliptic divisibility sequences arise this way.
• If an elliptic curve E is given in the Weierstrass form y2 = x3 + Ax + B over some field K, i.e. $A, B\in K$ one can evaluate the division polynomials at those A,B and consider them as polynomials in the coordinate ring. Then the powers of y can only be less or equal to 1, as y2 is replaced by x3 + Ax + B. In particular, ψ2n + 1 is a univariate polynomial in x only. The roots of the (2n + 1)th division polynomial ψ2n + 1 are exactly the x coordinates of the points of $E[2n+1]\setminus \{O\}$, where E[2n + 1] is the (2n + 1)th torsion subgroup of the group E of an elliptic curve.
• Given a point P = (xP,yP) on the elliptic curve E:y2 = x3 + 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 ϕn and ωn 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 ψ2m and ψ2m + 1, along with the equation of the curve, we have that $\psi_{n}^{2}$ , $\frac{\psi_{2n}}{y}, \psi_{2n + 1}$ and ϕn are all functions in the variable x.

Let p > 3 be prime and let $E:y^2=x^3+Ax+B, A,B \in \mathbb{F}_p$ be an elliptic curve over the finite field $\in \mathbb{F}_p$. The l-torsion group of E over $\bar{ \mathbb{F}}_p$ is isomorphic to $\mathbb{Z}/l \times \mathbb{Z}/l$ if $l\neq p$ and to $\mathbb{Z}/l$ of {0} otherwise which means that the degree of ψl is (l2 − 1) / 2, (l − 1) / 2 or 0.

René Schoof observed that working modulo the lth division polynomial means working with all l-torsion points simultaneously. This is heavily used in Schoof's algorithm for counting points on elliptic curves.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• División polinomial — Saltar a navegación, búsqueda En álgebra, división polinomial es un algoritmo que permite dividir un polinomio por otro polinomio de igual o menor grado. El algoritmo es una versión generalizada de la técnica aritmética de división. Es fácilmente …   Wikipedia Español

• Division (mathematics) — Divided redirects here. For other uses, see Divided (disambiguation). For the digital implementation of mathematical division, see Division (digital). In mathematics, especially in elementary arithmetic, division (÷ …   Wikipedia

• Greatest common divisor of two polynomials — Informally, the greatest common divisor (GCD) of two polynomials p ( x ) and q ( x ) is the biggest polynomial that divides evenly into both p ( x ) and q ( x ). The definition is modeled on the concept of the greatest common divisor of two… …   Wikipedia

• Polynomial long division — In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalised version of the familiar arithmetic technique called long division. It can be done easily by hand,… …   Wikipedia

• Chebyshev polynomials — Not to be confused with discrete Chebyshev polynomials. In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre s formula and which can be defined… …   Wikipedia

• Multivariate division algorithm — In mathematics, polynomials in more than one variable do not form a Euclidean domain, so it is not possible to construct a true division algorithm; but an approximate multivariate division algorithm can be constructed. Given a polynomial g,… …   Wikipedia

• Calculus with polynomials — Topics in Calculus Fundamental theorem Limits of functions Continuity Mean value theorem Differential calculus  Derivative Change of variables Implicit differentiation Taylor s theorem Related rates …   Wikipedia

• Long division — For the album by Rustic Overtones, see Long Division.In arithmetic, long division is the standard procedure suitable for dividing simple or complex multidigit numbers. It breaks down a division problem into a series of easier steps. As in all… …   Wikipedia

• Counting points on elliptic curves — An important aspect in the study of elliptic curves is devising effective ways of counting points on the curve. There have been several approaches to do so, and the algorithms devised have proved to be useful tools in the study of various fields… …   Wikipedia

• Riemann surface — For the Riemann surface of a subring of a field, see Zariski–Riemann space. Riemann surface for the function ƒ(z) = √z. The two horizontal axes represent the real and imaginary parts of z, while the vertical axis represents the real… …   Wikipedia