Explicit-Formulas Database
Ordinary genus-1 curves over binary fields

# Binary Edwards curves

An elliptic curve in binary Edwards form [database entry; Sage verification script; 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)
```
```  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

```  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
```

```  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
```

```  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
```

```  d1=d2
```  x=X/Z
```  x=X/Z