 MPHELL  4.0.0
Jacobi Quartic elliptic curves

# Jacobi Quartic elliptic curves

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.

# Extended Jacobi Quartic elliptic curves with d = 1

These formulas are derived from the paper [HKCD09] "Jacobi Quartic Curves Revisited" written by Huseyin Hisil, Kenneth Koon-Ho Wong, Gary Carter and Ed Dawson published in Australasian Conference on Information Security and Privacy 2009 (pp. 452-468). 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

1. 2. The point is given by

1. 2. 3. 4. 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].

## Unified Doubling

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.

## Dedicated Doubling

Let

1. The point is given by

1. 2. 3. 4. using 3 multiplications, 5 squares, 1 times a, 13 additions and 4 times 2

More details are in [HKCD09,§3.1].