Explicit-Formulas Database

Genus-1 curves over large-characteristic fields
# Jacobi intersections

An elliptic curve in Jacobi intersection form
[database entry;
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has parameters
a
and coordinates
s
c
d
satisfying the following equations:
s^^{2}+c^^{2}=1
a*s^^{2}+d^^{2}=1

Affine addition formulas: (s1,c1,d1)+(s2,c2,d2)=(s3,c3,d3) where
s3 = (c2*s1*d2+d1*s2*c1)/(c2^^{2}+(d1*s2)^^{2})
c3 = (c2*c1-d1*s2*s1*d2)/(c2^^{2}+(d1*s2)^^{2})
d3 = (d1*d2-a*s1*c1*s2*c2)/(c2^^{2}+(d1*s2)^^{2})

Affine doubling formulas: 2(s1,c1,d1)=(s3,c3,d3) where
s3 = (c1*s1*d1+d1*s1*c1)/(c1^^{2}+(d1*s1)^^{2})
c3 = (c1*c1-d1*s1*s1*d1)/(c1^^{2}+(d1*s1)^^{2})
d3 = (d1*d1-a*s1*c1*s1*c1)/(c1^^{2}+(d1*s1)^^{2})

Affine negation formulas: -(s1,c1,d1)=(-s1,c1,d1).
The neutral element of a Jacobi intersection is the point (0,1,1).
The parameter a is required to be different from 0 and 1.

## Representations for fast computations

Extended coordinates
[more information]
represent
s
c
d
as
S
C
D
Z
SC
DZ
satisfying the following equations:
s=S/Z
c=C/Z
d=D/Z
SC=S*C
DZ=D*Z

Projective coordinates
[more information]
represent
s
c
d
as
S
C
D
Z
satisfying the following equations:

s=S/Z
c=C/Z
d=D/Z