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)
```
```  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
```

```  c=1
d=r^2
```
and represent x y as Y Z satisfying the following equations:
```  r*y=Y/Z
```

```  c=1
d=r^2
```
and represent x y as Y Z satisfying the following equations:
```  r*y^2=Y/Z
```