Explicit-Formulas Database

Genus-1 curves over large-characteristic fields
# Short Weierstrass curves

An elliptic curve in short Weierstrass form
[database entry;
Sage verification script;
Sage output]
has parameters
a
b
and coordinates
x
y
satisfying the following equations:
y^^{2}=x^^{3}+a*x+b

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

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

Affine negation formulas: -(x1,y1)=(x1,-y1).
The neutral element of the curve is the unique point at infinity,
namely (0:1:0) in projective coordinates.

## Representations for fast computations

Jacobian coordinates with a4=0
[more information]
make the additional assumptions
a=0

and
represent
x
y
as
X
Y
Z
satisfying the following equations:
x=X/Z^^{2}
y=Y/Z^^{3}

Jacobian coordinates with a4=-3
[more information]
make the additional assumptions

a=-3

and
represent
x
y
as
X
Y
Z
satisfying the following equations:
x=X/Z^^{2}
y=Y/Z^^{3}

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

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

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

x=X/Z^^{2}
y=Y/Z^^{3}
T=a*Z^^{4}

Projective coordinates with a4=-1
[more information]
make the additional assumptions

a=-1

and
represent
x
y
as
X
Y
Z
satisfying the following equations:
x=X/Z
y=Y/Z

Projective coordinates with a4=-3
[more information]
make the additional assumptions

a=-3

and
represent
x
y
as
X
Y
Z
satisfying the following equations:
x=X/Z
y=Y/Z

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

x=X/Z
y=Y/Z

W12 coordinates with a6=0
[more information]
make the additional assumptions

b=0

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

XYZZ coordinates with a4=-3
[more information]
make the additional assumptions

a=-3

and
represent
x
y
as
X
Y
ZZ
ZZZ
satisfying the following equations:
x=X/ZZ
y=Y/ZZZ
ZZ^^{3}=ZZZ^^{2}

XYZZ coordinates
[more information]
represent
x
y
as
X
Y
ZZ
ZZZ
satisfying the following equations:

x=X/ZZ
y=Y/ZZZ
ZZ^^{3}=ZZZ^^{2}

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

x=X/Z