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

${\displaystyle \psi _{0}=0}$ 

${\displaystyle \psi _{1}=1}$ 

${\displaystyle \psi _{2}=2y}$ 

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

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

${\displaystyle \vdots }$ 

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

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

The polynomial ${\displaystyle \psi _{n}}$  is called the n^th division polynomial.

Properties

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

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

where ${\displaystyle \phi _{n}}$  and ${\displaystyle \omega _{n}}$  are defined by:

${\displaystyle \phi _{n}=x\psi _{n}^{2}-\psi _{n+1}\psi _{n-1},}$ 

${\displaystyle \omega _{n}={\frac {\psi _{n+2}\psi _{n-1}^{2}-\psi _{n-2}\psi _{n+1}^{2}}{4y}}.}$ 

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

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

René Schoof observed that working modulo the ${\displaystyle \ell }$ th division polynomial allows one to work with all ${\displaystyle \ell }$ -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 ${\displaystyle E(\mathbb {F} _{q})}$ . 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.