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