Twisted Edwards curves

An elliptic curve in twisted Edwards form [database entry; Sage verification script; Sage output] has parameters a d and coordinates x y satisfying the following equations:

  a*x^²+y^²=1+d*x^²*y^²

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

  x3 = (x1*y2+y1*x2)/(1+d*x1*x2*y1*y2)
  y3 = (y1*y2-a*x1*x2)/(1-d*x1*x2*y1*y2)

Affine doubling formulas: 2(x1,y1)=(x3,y3) where

  x3 = (x1*y1+y1*x1)/(1+d*x1*x1*y1*y1)
  y3 = (y1*y1-a*x1*x1)/(1-d*x1*x1*y1*y1)

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

Representations for fast computations

Extended coordinates with a=-1 [more information] make the additional assumptions

  a=-1

and represent x y as X Y Z T satisfying the following equations:

  x=X/Z
  y=Y/Z
  x*y=T/Z

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

  x=X/Z
  y=Y/Z
  x*y=T/Z

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

  x=Z/X
  y=Z/Y

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

  x=X/Z
  y=Y/Z