MPHELL
4.0.0

Jacobi Quartics elliptic curves are elliptic curves given by the equation
Let be a Jacobi Quartic elliptic curve under affine coordinates.
Under projective coordinates becomes . The triplet (X, Y, Z) represents the affine point (X / Z, Y / Z).
Under extended homogenous projective coordinates becomes . The quadruplet (X, Y, T, Z) represents the affine point (X / Z, Y / Z) with .
An elliptic curve with this equation is called an extended jacobi quartic elliptic curve, in MPHELL we work only in this case with in the extended homogeneous projective coordinates.
These formulas are derived from the paper [HKCD09] "Jacobi Quartic Curves Revisited" written by Huseyin Hisil, Kenneth KoonHo Wong, Gary Carter and Ed Dawson published in Australasian Conference on Information Security and Privacy 2009 (pp. 452468). Springer, Berlin, Heidelberg viewable at this adress : https://link.springer.com/ .
Let a Jacobi Quartic elliptic curve under extended homogeneous projective coordinates.
Under affine coordinates becomes .
Let
The point is given by
using 8 multiplications, 2 squares and 1 times a 13 additions and 1 times 2.
It is noticeable that this is a simplified formula of a more general formula for extended jacobi quartic.
More details are in [HKCD09,§4.3,B].
With use the unified addition:
We use the unified addition formulae because it is in the general case the fastest addition in extended homogeneous projective coordinates.
Let
The point is given by
using 3 multiplications, 5 squares, 1 times a, 13 additions and 4 times 2
More details are in [HKCD09,§3.1].