Edwards curves

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

 x^²+y^²=c^²*(1+d*x^²*y^²)

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

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

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

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

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

This curve shape was introduced by 2007 Edwards for the case d=1.

Technically, an Edwards curve is not elliptic, because it has singularities; but resolving those singularities produces an elliptic curve.

The neutral element of the curve is the point (0,c). The point (0,-c) has order 2. The points (c,0) and (-c,0) have order 4.

Representations for fast computations

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

YZ coordinates with square d [more information] make the additional assumptions

 c=1
 d=r^²

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

 r*y=Y/Z

Squared YZ coordinates with square d [more information] make the additional assumptions

 c=1
 d=r^²

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

 r*y^²=Y/Z