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

Doubling-oriented Doche–Icart–Kohel curves

An elliptic curve in doubling-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+a*x^²+16*a*x

Affine addition formulas: (x1,y1)+(x2,y2)=(x3,y3) where

  x3 = (-x1^³+(x2-a)*x1^²+(x2^²+2*a*x2)*x1+(y1^²-2*y2*y1+(-x2^³-a*x2^²+y2^²)))/(x1^²-2*x2*x1+x2^²)
  y3 = ((-y1+2*y2)*x1^³+(-a*y1+(-3*y2*x2+a*y2))*x1^²+((3*x2^²+2*a*x2)*y1-2*a*y2*x2)*x1+(y1^³-3*y2*y1^²+(-2*x2^³-a*x2^²+3*y2^²)*y1+(y2*x2^³+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 = 1/(4*y1^²)*x1^⁴-8*a/y1^²*x1^²+64*a^²/y1^²
  y3 = 1/(8*y1^³)*x1^⁶+((-a^²+40*a)/(4*y1^³))*x1^⁴+((a*y1^²+(16*a^³-640*a^²))/(4*y1^³))*x1^²+((-4*a^²*y1^²-512*a^³)/y1^³)

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

This curve shape was introduced in 2006 Doche–Icart–Kohel. The parameter a is required to have a(a-64) nonzero. The neutral element of the curve is the unique point at infinity, namely (0:1:0) in projective coordinates.

Representations for fast computations

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

  x=X/Z
  y=Y/ZZ
  ZZ=Z^²