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

Binary Edwards curves

An elliptic curve in binary Edwards form [database entry; Sage verification script; Sage output] has parameters d1 d2 and coordinates x y satisfying the following equations:

  d1*(x+y)+d2*(x^²+y^²)=(x+x^²)*(y+y^²)

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

  x3 = (d1*(x1+x2)+d2*(x1+y1)*(x2+y2)+(x1+x1^²)*(x2*(y1+y2+1)+y1*y2))/(d1+(x1+x1^²)*(x2+y2))
  y3 = (d1*(y1+y2)+d2*(x1+y1)*(x2+y2)+(y1+y1^²)*(y2*(x1+x2+1)+x1*x2))/(d1+(y1+y1^²)*(x2+y2))

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

  x3 = (d1*(x1+x1)+d2*(x1+y1)*(x1+y1)+(x1+x1^²)*(x1*(y1+y1+1)+y1*y1))/(d1+(x1+x1^²)*(x1+y1))
  y3 = (d1*(y1+y1)+d2*(x1+y1)*(x1+y1)+(y1+y1^²)*(y1*(x1+x1+1)+x1*x1))/(d1+(y1+y1^²)*(x1+y1))

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

Representations for fast computations

W coordinates with d1=d2 [more information] make the additional assumptions

  d1=d2

and represent x y as w satisfying the following equations:

  x+y=w

W coordinates [more information] represent x y as w satisfying the following equations:

  x+y=w

WZ coordinates with d1=d2 [more information] make the additional assumptions

  d1=d2

and represent x y as W Z satisfying the following equations:

  x+y=W/Z

WZ coordinates [more information] represent x y as W Z satisfying the following equations:

  x+y=W/Z

Affine coordinates with d1=d2 [more information] make the additional assumptions

  d1=d2

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

  x=X
  y=Y

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

  x=X
  y=Y

Projective coordinates with d1=d2 [more information] make the additional assumptions

  d1=d2

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

  x=X/Z
  y=Y/Z

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

  x=X/Z
  y=Y/Z