Explicit-Formulas Database
Genus-1 curves over large-characteristic 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

Affine addition formulas: (x1,y1)+(x2,y2)=(x3,y3) where

 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:-w^²:0).

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

Extended coordinates [more information] represent x y as X Y Z XX YY ZZ XY YZ XZ satisfying the following equations:

 x=X/Z
 y=Y/Z
 XX=X*X
 YY=Y*Y
 ZZ=Z*Z
 XY=2*X*Y
 XZ=2*X*Z
 YZ=2*Y*Z

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

 x=X/Z
 y=Y/Z