Explicit-Formulas Database
Genus-1 curves over large-characteristic fields
Tripling-oriented Doche–Icart–Kohel curves
An elliptic curve in tripling-oriented Doche–Icart–Kohel form
[database entry;
Sage verification script;
Sage output]
has parameters
a
and coordinates
x
y
satisfying the following equations:
y^2=x^3+3*a*(x+1)^2
Tripling-oriented Doche–Icart–Kohel curves
were introduced in 2006 Doche–Icart–Kohel.
The neutral element of the curve is the unique point at infinity,
namely (0:1:0) in projective coordinates.
The parameter a is required to be different from 0 and 9/4.
Representations for fast computations
Standard coordinates
[more information]
represent
x
y
as
X
Y
Z
ZZ
satisfying the following equations:
x=X/Z^2
y=Y/Z^3
ZZ=Z^2