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

# Hessian curves

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

The neutral element of a Hessian curve is a point at infinity, namely (1:-1:0) in projective coordinates. Over a field with a nontrivial cube root w of 1 there are two other points at infinity, namely (1:-w:0) and (1:-w2:0).

2001 Joye Quisquater state a birational equivalence between a Hessian curve with neutral element (-1:0:1) and a Weierstrass curve with neutral element at infinity. EFD permutes coordinates to obtain a birational equivalence between a Hessian curve with neutral element (1:-1:0) and a Weierstrass curve with neutral element at infinity.

## Representations for fast computations

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