Division polynomials

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 \ellth 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.