Twisted Edwards Curves
The use of Edwards curves needs fewer modular multiplications and less time than the use of Montgomery curves or Weierstrass curves (other used methods). Using Edwards curves you can also find more primes.
Definition: Let be a field in which, and let with . Then the twisted Edwards curve is given by An Edwards curve is a twisted Edwards curve in which .
There are five known ways to build a set of point on an Edwards curve: the set of affine points, the set of projective points, the set of inverted points, the set of extended points and the set of completed points.
The set of affine points is given by: .
The addition law is given by . The point (0,1) is its neutral element and the negative of is . The other representations are defined similar to how the projective Weierstrass curve follows from the affine.
Any elliptic curve in Edwards form has a point of order 4. So the torsion group of an Edwards curve over is isomorphic to either or .
The most interesting cases for ECM are and, since they force the group orders of the curve modulo primes to be divisible by 12 and 16 respectively. The following curves have a torsion group isomorphic to :
- with point where and
- with point where and
Every Edwards curve with a point of order 3 can be written in the ways shown above. Curves with torsion group isomorphic to and can be found on http://eprint.iacr.org/2008/016 page 30-32.
Read more about this topic: Lenstra Elliptic Curve Factorization
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