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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 $ E_w: y^2 = x^3 + a_wx + b_w $ a Weierstrass elliptic curve.

Let $ E_j: y^2 = z^2 + 2a_jx^2 + b_jt^2; x^2 = zt; 256b_j(a_j^2-b_j^2) \ne 0 $ an extended Jacobi Quartic elliptic curve.

For $ z = 1; E_j: y^2 = b_j x^4 + 2a_jx^2 + 1 $ which can be extented to the Jacobi Quartic (with Projective equation) $ E_{jext}: y^2 z^2 = b_jx^4 + 2a_jx^2z^2 + z^4 $

Jacobi Quartic -> Weierstrass

Curve conversion

The coefficients $ a_w $ and $ b_w $ of the Weiertrass elliptic curve $ E_w $ matching the Jacobi Quartic elliptic curve $ E_j $ are:

  1. $ a_w = -4((a_j^2)/3 + b_j) $
  2. $ b_w = (64/27).a_j^3 + (4/3)a_j.a_w $

Point conversion

A point $ (X,Y,Z) $ of $ E_{jext} $ can be converted to a point $ (x,y) $ of $ E_w $ with

  1. $ x = 2.(Y+Z^2)/X^2 + 2.a_j/3 $
  2. $ y = Z.(4.(Y + Z^2) + 4.a_j.X^2)/X^3 $

Weierstrass -> Jacobi Quartic

Let $\theta$ such that $(\theta, 0)$ is a 2-torsion point the Weierstrass elliptic curve $ E_w $.

Curve conversion

The coefficients $ b_j $ and $ a_j $ of the (extended) Jacobi Quartic $ E_{jext} $ matching the Weiertrass elliptic curve $ E_w $ are:

  1. $ b_j = -(3.\theta^2 + 4.a_w)/16 $
  2. $ a_j = -(3.\theta)/4 $

Point conversion

A point $ (x,y) $ of $ E_w $ can be converted to a point $ (X,Y,Z,T) $ of $ E_j $ with

  1. $ X = (2.x + (8/3).a_j).y $
  2. $ Y = (2.x - (4/3).a_j)(x + (4/3).a_j)^2 -y^2$
  3. $ Z = y^2 $
  4. $ T = X^2 / Z $

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

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