Explicit-Formulas Database

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

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

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

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

Affine negation formulas: -(x1,y1)=(-x1,y1).
This curve shape was introduced by 2007 Edwards for the case d=1.

Technically, an Edwards curve is not elliptic, because it has singularities;
but resolving those singularities produces an elliptic curve.

The neutral element of the curve is the point (0,c).
The point (0,-c) has order 2.
The points (c,0) and (-c,0) have order 4.

## Representations for fast computations

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

YZ coordinates with square d
[more information]
make the additional assumptions

c=1
d=r^^{2}

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

Squared YZ coordinates with square d
[more information]
make the additional assumptions

c=1
d=r^^{2}

and
represent
x
y
as
Y
Z
satisfying the following equations:
r*y^^{2}=Y/Z