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)

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