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.

Contents

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.

See also

References

  • A. Brown: Algorithms for Elliptic Curves over Finite Fields, EPFL — LMA. Available at http://algo.epfl.ch/handouts/en/andrew.pdf
  • 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 kurvenüber 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 http://www-math.mit.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.

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