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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Sage output]
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