Explicit-Formulas Database

Genus-1 curves over large-characteristic fields
# Twisted Hessian curves

An elliptic curve in twisted Hessian form
[database entry;
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has parameters
a
d
and coordinates
x
y
satisfying the following equations:
a*x^^{3}+y^^{3}+1=d*x*y

Affine addition formulas: (x1,y1)+(x2,y2)=(x3,y3) where
x3 = (x1-y1^^{2}*x2*y2)/(a*x1*y1*x2^^{2}-y2)
y3 = (y1*y2^^{2}-a*x1^^{2}*x2)/(a*x1*y1*x2^^{2}-y2)

Affine doubling formulas: 2(x1,y1)=(x3,y3) where
x3 = (x1-y1^^{3}*x1)/(a*y1*x1^^{3}-y1)
y3 = (y1^^{3}-a*x1^^{3})/(a*y1*x1^^{3}-y1)

Affine negation formulas: -(x1,y1)=(x1/y1,1/y1).

## 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