Division polynomials

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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

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The set of division polynomials is a sequence of polynomials in   with   free variables that is recursively defined by:

 
 
 
 
 
 
 
 

The polynomial   is called the nth division polynomial.

Properties

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  • In practice, one sets  , and then   and  .
  • The division polynomials form a generic elliptic divisibility sequence over the ring  .
  • If an elliptic curve   is given in the Weierstrass form   over some field  , i.e.  , one can use these values of   and consider the division polynomials in the coordinate ring of  . The roots of   are the  -coordinates of the points of  , where   is the   torsion subgroup of  . Similarly, the roots of   are the  -coordinates of the points of  .
  • Given a point   on the elliptic curve   over some field  , we can express the coordinates of the nth multiple of   in terms of division polynomials:
 
where   and   are defined by:
 
 

Using the relation between   and  , along with the equation of the curve, the functions   ,  ,   are all in  .

Let   be prime and let   be an elliptic curve over the finite field  , i.e.,  . The  -torsion group of   over   is isomorphic to   if  , and to   or   if  . Hence the degree of   is equal to either  ,  , or 0.

René Schoof observed that working modulo the  th division polynomial allows one to work with all  -torsion points simultaneously. This is heavily used in Schoof's algorithm for counting points on elliptic curves.

See also

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References

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  • 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  . 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.