Explicit-Formulas Database

Genus-1 curves over large-characteristic fields
# Twisted Edwards curves

An elliptic curve in twisted Edwards form
[database entry;
Sage verification script;
Sage output]
has parameters
a
d
and coordinates
x
y
satisfying the following equations:
a*x^^{2}+y^^{2}=1+d*x^^{2}*y^^{2}

Affine addition formulas: (x1,y1)+(x2,y2)=(x3,y3) where
x3 = (x1*y2+y1*x2)/(1+d*x1*x2*y1*y2)
y3 = (y1*y2-a*x1*x2)/(1-d*x1*x2*y1*y2)

Affine doubling formulas: 2(x1,y1)=(x3,y3) where
x3 = (x1*y1+y1*x1)/(1+d*x1*x1*y1*y1)
y3 = (y1*y1-a*x1*x1)/(1-d*x1*x1*y1*y1)

Affine negation formulas: -(x1,y1)=(-x1,y1).

## Representations for fast computations

Extended coordinates with a=-1
[more information]
make the additional assumptions
a=-1

and
represent
x
y
as
X
Y
Z
T
satisfying the following equations:
x=X/Z
y=Y/Z
x*y=T/Z

Extended coordinates
[more information]
represent
x
y
as
X
Y
Z
T
satisfying the following equations:

x=X/Z
y=Y/Z
x*y=T/Z

Inverted coordinates
[more information]
represent
x
y
as
X
Y
Z
satisfying the following equations:

x=Z/X
y=Z/Y

Projective coordinates
[more information]
represent
x
y
as
X
Y
Z
satisfying the following equations:

x=X/Z
y=Y/Z