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
```
```  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

```  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
```