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

# Tripling-oriented Doche–Icart–Kohel curves

An elliptic curve in tripling-oriented Doche–Icart–Kohel form [database entry; Sage verification script; Sage output] has parameters a and coordinates x y satisfying the following equations:
```  y^2=x^3+3*a*(x+1)^2
```
Affine addition formulas: (x1,y1)+(x2,y2)=(x3,y3) where
```  x3 = (-x1^3+(x2-3*a)*x1^2+(x2^2+6*a*x2)*x1+(y1^2-2*y2*y1+(-x2^3-3*a*x2^2+y2^2)))/(x1^2-2*x2*x1+x2^2)
y3 = ((-y1+2*y2)*x1^3+(-3*a*y1+(-3*y2*x2+3*a*y2))*x1^2+((3*x2^2+6*a*x2)*y1-6*a*y2*x2)*x1+(y1^3-3*y2*y1^2+(-2*x2^3-3*a*x2^2+3*y2^2)*y1+(y2*x2^3+3*a*y2*x2^2-y2^3)))/(-x1^3+3*x2*x1^2-3*x2^2*x1+x2^3)
```
Affine doubling formulas: 2(x1,y1)=(x3,y3) where
```  x3 = 9/(4*y1^2)*x1^4+9/y1^2*a*x1^3+(9/y1^2*a^2+9/y1^2*a)*x1^2+(18/y1^2*a^2-2)*x1+(9/y1^2*a^2-3*a)
y3 = -27/(8*y1^3)*x1^6-81/(4*y1^3)*a*x1^5+(-81/(2*y1^3)*a^2-81/(4*y1^3)*a)*x1^4+(-27/y1^3*a^3-81/y1^3*a^2+9/(2*y1))*x1^3+(-81/y1^3*a^3-81/(2*y1^3)*a^2+27/(2*y1)*a)*x1^2+(-81/y1^3*a^3+9/y1*a^2+9/y1*a)*x1+(-27/y1^3*a^3+9/y1*a^2-y1)
```
Affine negation formulas: -(x1,y1)=(x1,-y1).

Tripling-oriented Doche–Icart–Kohel curves were introduced in 2006 Doche–Icart–Kohel.

The neutral element of the curve is the unique point at infinity, namely (0:1:0) in projective coordinates. The parameter a is required to be different from 0 and 9/4.

## Representations for fast computations

Standard coordinates [more information] represent x y as X Y Z ZZ satisfying the following equations:
```  x=X/Z^2
y=Y/Z^3
ZZ=Z^2
```