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

# Short Weierstrass curves

An elliptic curve in short Weierstrass form [database entry; Sage verification script; Sage output] has parameters a2 a6 and coordinates x y satisfying the following equations:
```  y^2+x*y=x^3+a2*x^2+a6
```
```  x3 = ((y1+y2)/(x1+x2))^2+((y1+y2)/(x1+x2))+x1+x2+a2
y3 = ((y1+y2)/(x1+x2))^3+(x2+a2+1)*((y1+y2)/(x1+x2))+x1+x2+a2+y1
```
Affine doubling formulas: 2(x1,y1)=(x3,y3) where
```  x3 = (x1+y1/x1)^2+(x1+y1/x1)+a2
y3 = (x1+y1/x1)^3+(x1+a2+1)*(x1+y1/x1)+a2+y1
```
Affine negation formulas: -(x1,y1)=(x1,x1+y1).

## Representations for fast computations

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

```  a2=0
```
and represent x y as X Y Z ZZ XZ satisfying the following equations:
```  x=X/Z
y=Y/ZZ
ZZ=Z^2
XZ=X*Z
```

```  a2=1
```
and represent x y as X Y Z ZZ satisfying the following equations:
```  x=X/Z
y=Y/Z^2
ZZ=Z^2
```

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

```  x=X/Z^2
y=Y/Z^3
```

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

```  x=X/Z
y/x=(L-X)/Z
```

```  a2=0
```
and represent x y as X Y Z satisfying the following equations:
```  x=X/Z
y=Y/Z^2
```

```  a2=1
```
and represent x y as X Y Z satisfying the following equations:
```  x=X/Z
y=Y/Z^2
```

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

```  x=X/Z
y=Y/Z^2
```

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

```  x=X/Z
y=Y/Z
```

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

```  x=X/Z
```