Explicit-Formulas Database
Genus-1 curves over large-characteristic fields

Jacobi intersections

An elliptic curve in Jacobi intersection form [database entry; Sage verification script; Sage output] 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