 MPHELL  4.0.0
Conversion between Weiestrass and Jacobi Quartic elliptic curves

These formulas are derived from the paper [BJ03] "The Jacobi Model of an Elliptic Curve and Side-Channel Analysis" written by Olivier Billet and Marc Joye published in International Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes. Springer, Berlin, Heidelberg, 2003. p. 34-42 viewable at this address https://link.springer.com/ .

Let a Weierstrass elliptic curve.

Let an extended Jacobi Quartic elliptic curve.

For which can be extented to the Jacobi Quartic (with Projective equation) # Jacobi Quartic -> Weierstrass

## Curve conversion

The coefficients and of the Weiertrass elliptic curve matching the Jacobi Quartic elliptic curve are:

1. 2. ## Point conversion

A point of can be converted to a point of with

1. 2. # Weierstrass -> Jacobi Quartic

Let such that is a 2-torsion point the Weierstrass elliptic curve .

## Curve conversion

The coefficients and of the (extended) Jacobi Quartic matching the Weiertrass elliptic curve are:

1. 2. ## Point conversion

A point of can be converted to a point of with

1. 2. 3. 4. Those formulas are corrected formulas of [BJ03,§3]. More details are available in [BJ03,§3].