Explicit-Formulas Database

Ordinary genus-1 curves over binary fields
# Short Weierstrass curves

An elliptic curve in short Weierstrass form
[database entry;
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has parameters
a2
a6
and coordinates
x
y
satisfying the following equations:
y^^{2}+x*y=x^^{3}+a2*x^^{2}+a6

Affine addition formulas: (x1,y1)+(x2,y2)=(x3,y3) where
x3 = ((y1+y2)/(x1+x2))^^{2}+((y1+y2)/(x1+x2))+x1+x2+a2
y3 = ((y1+y2)/(x1+x2))^^{3}+(x2+a2+1)*((y1+y2)/(x1+x2))+x1+x2+a2+y1

Affine doubling formulas: 2(x1,y1)=(x3,y3) where
x3 = (x1+y1/x1)^^{2}+(x1+y1/x1)+a2
y3 = (x1+y1/x1)^^{3}+(x1+a2+1)*(x1+y1/x1)+a2+y1

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

## Representations for fast computations

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

Extended Lopez-Dahab coordinates with a2=0
[more information]
make the additional assumptions

a2=0

and
represent
x
y
as
X
Y
Z
ZZ
XZ
satisfying the following equations:
x=X/Z
y=Y/ZZ
ZZ=Z^^{2}
XZ=X*Z

Extended Lopez-Dahab coordinates with a2=1
[more information]
make the additional assumptions

a2=1

and
represent
x
y
as
X
Y
Z
ZZ
satisfying the following equations:
x=X/Z
y=Y/Z^^{2}
ZZ=Z^^{2}

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

x=X/Z^^{2}
y=Y/Z^^{3}

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

x=X/Z
y/x=(L-X)/Z

Lopez-Dahab coordinates with a2=0
[more information]
make the additional assumptions

a2=0

and
represent
x
y
as
X
Y
Z
satisfying the following equations:
x=X/Z
y=Y/Z^^{2}

Lopez-Dahab coordinates with a2=1
[more information]
make the additional assumptions

a2=1

and
represent
x
y
as
X
Y
Z
satisfying the following equations:
x=X/Z
y=Y/Z^^{2}

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

x=X/Z
y=Y/Z^^{2}

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

x=X/Z
y=Y/Z

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

x=X/Z