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.
The set of division polynomials is a sequence of polynomials in
free variables that is recursively defined by: The polynomial
ψ
is called the nth division polynomial.
ψ
ψ
, along with the equation of the curve, the functions
ψ
be an elliptic curve over the finite field
ℓ
ℓ ×
ℓ
ℓ ≠ p
ℓ
ℓ = p
ℓ
René Schoof observed that working modulo the
ℓ
th division polynomial allows one to work with all
ℓ
-torsion points simultaneously.
This is heavily used in Schoof's algorithm for counting points on elliptic curves.