Explicit-Formulas Database

Ordinary genus-1 curves over binary fields
# Binary Edwards curves

An elliptic curve in binary Edwards form
[database entry;
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Sage output]
has parameters
d1
d2
and coordinates
x
y
satisfying the following equations:
d1*(x+y)+d2*(x^^{2}+y^^{2})=(x+x^^{2})*(y+y^^{2})

Affine addition formulas: (x1,y1)+(x2,y2)=(x3,y3) where
x3 = (d1*(x1+x2)+d2*(x1+y1)*(x2+y2)+(x1+x1^^{2})*(x2*(y1+y2+1)+y1*y2))/(d1+(x1+x1^^{2})*(x2+y2))
y3 = (d1*(y1+y2)+d2*(x1+y1)*(x2+y2)+(y1+y1^^{2})*(y2*(x1+x2+1)+x1*x2))/(d1+(y1+y1^^{2})*(x2+y2))

Affine doubling formulas: 2(x1,y1)=(x3,y3) where
x3 = (d1*(x1+x1)+d2*(x1+y1)*(x1+y1)+(x1+x1^^{2})*(x1*(y1+y1+1)+y1*y1))/(d1+(x1+x1^^{2})*(x1+y1))
y3 = (d1*(y1+y1)+d2*(x1+y1)*(x1+y1)+(y1+y1^^{2})*(y1*(x1+x1+1)+x1*x1))/(d1+(y1+y1^^{2})*(x1+y1))

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

## Representations for fast computations

W coordinates with d1=d2
[more information]
make the additional assumptions
d1=d2

and
represent
x
y
as
w
satisfying the following equations:
x+y=w

W coordinates
[more information]
represent
x
y
as
w
satisfying the following equations:

x+y=w

WZ coordinates with d1=d2
[more information]
make the additional assumptions

d1=d2

and
represent
x
y
as
W
Z
satisfying the following equations:
x+y=W/Z

WZ coordinates
[more information]
represent
x
y
as
W
Z
satisfying the following equations:

x+y=W/Z

Affine coordinates with d1=d2
[more information]
make the additional assumptions

d1=d2

and
represent
x
y
as
X
Y
satisfying the following equations:
x=X
y=Y

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

x=X
y=Y

Projective coordinates with d1=d2
[more information]
make the additional assumptions

d1=d2

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

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

x=X/Z
y=Y/Z