Definition
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
- 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.
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