Explicit-Formulas Database
Ordinary genus-1 curves over binary fields

Short Weierstrass curves

An elliptic curve in short Weierstrass form [database entry; Sage verification script; Sage output] has parameters a2 a6 and coordinates x y satisfying the following equations:
  y2+x*y=x3+a2*x2+a6
Affine addition formulas: (x1,y1)+(x2,y2)=(x3,y3) where
  x3 = ((y1+y2)/(x1+x2))2+((y1+y2)/(x1+x2))+x1+x2+a2
  y3 = ((y1+y2)/(x1+x2))3+(x2+a2+1)*((y1+y2)/(x1+x2))+x1+x2+a2+y1
Affine doubling formulas: 2(x1,y1)=(x3,y3) where
  x3 = (x1+y1/x1)2+(x1+y1/x1)+a2
  y3 = (x1+y1/x1)3+(x1+a2+1)*(x1+y1/x1)+a2+y1
Affine negation formulas: -(x1,y1)=(x1,x1+y1).

Representations for fast computations

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

Extended Lopez-Dahab coordinates with a2=0 [more information] make the additional assumptions

  a2=0
and represent x y as X Y Z ZZ XZ satisfying the following equations:
  x=X/Z
  y=Y/ZZ
  ZZ=Z2
  XZ=X*Z

Extended Lopez-Dahab coordinates with a2=1 [more information] make the additional assumptions

  a2=1
and represent x y as X Y Z ZZ satisfying the following equations:
  x=X/Z
  y=Y/Z2
  ZZ=Z2

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

  x=X/Z2
  y=Y/Z3

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

  x=X/Z
  y/x=(L-X)/Z

Lopez-Dahab coordinates with a2=0 [more information] make the additional assumptions

  a2=0
and represent x y as X Y Z satisfying the following equations:
  x=X/Z
  y=Y/Z2

Lopez-Dahab coordinates with a2=1 [more information] make the additional assumptions

  a2=1
and represent x y as X Y Z satisfying the following equations:
  x=X/Z
  y=Y/Z2

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

  x=X/Z
  y=Y/Z2

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

  x=X/Z
  y=Y/Z

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

  x=X/Z