MPHELL  5.0.0
Jacobi Quartic elliptic curves

Jacobi Quartic elliptic curves

Jacobi Quartics elliptic curves are elliptic curves given by the equation

Let $ E_j: y^2= d_j x^4 + 2 a_j x^2 +1, d_j = k^2 \textit{and } a_j=-(1+k^2)/2 $ be a Jacobi Quartic elliptic curve under affine coordinates.

Under projective coordinates $ E_j$ becomes $ Y^2Z^2= d_j X^4 + 2 a_j X^2 Z^2 + Z^4, d_j = k^2 \textit{and } a_j=-(1+k^2)/2 $. The triplet (X, Y, Z) represents the affine point (X / Z, Y / Z).

Under extended homogenous projective coordinates $ E_j $ becomes $ Y^2 = d_j T^2 + 2a_jX^2 + Z^2, d_j = k^2 \textit{, } a_j=-(1+k^2)/2 \textit{and } X^2=Z T$. The quadruplet (X, Y, T, Z) represents the affine point (X / Z, Y / Z) with $X^2=Z T$.

An elliptic curve $ E_j: y^2= d_j x^4 + 2 a_j x^2 +1 $ with this equation is called an extended jacobi quartic elliptic curve, in MPHELL we work only in this case with $ d_j=1 $ in the extended homogeneous projective coordinates.

Extended Jacobi Quartic elliptic curves with d = 1 <br>

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 $ E_j: Y^2 = T^2 + 2a_jX^2 + Z^2; X^2 = ZT; a_j \ne 1 $ a Jacobi Quartic elliptic curve under extended homogeneous projective coordinates.

Under affine coordinates $ E_j: y^2 = z^2 + 2a_jx^2 + t^2; x^2 = zt; a_j \ne 1 $ becomes $ E_j: y^2 = x^4 + 2a_jx^2 + 1;$.

Unified Addition

Let

  1. $ P1 = (X1, Y1, T1, Z1) $
  2. $ P2 = (X2, Y2, T2, Z2) $

The point $ P3 = (X3, Y3, T3, Z3) = P1 + P2 $ is given by

  1. $ X3 = (Z1 \times Z2 - T1 \times T2) \times (X1 \times Y2 + X2 \times Y1) $
  2. $ Y3 = (2 \times X1 \times X2 + Z1 \times Z2 + T1 \times T2) \times (Z1 \times T2 + Z2 \times T1 + Y1 \times Y2 + 2 \times X1 \times X2 \times a_j) - (X1 \times Y2 + X2 \times Y1)^2 $
  3. $ T3 = (X1 \times Y2 + X2 \times Y1)^2 $
  4. $ Z3 = (Z1 \times Z2 - T1 \times T2)^2 $

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:

$ 2 \times P1 = P1 + P1 $

Dedicated 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. $ P1 = (X1, Y1, T1, Z1) $

The point $ P3 = (X3, Y3, T3, Z3) = 2 P1 $ is given by

  1. $ X3 = 2 \times X1 \times Y1 \times (2 \times Z1^2 + 2 \times a \times X1^2 - Y1^2) $
  2. $ Y3 = 2 \times Y1^2 \times (Y1^2 - 2 \times a_j \times X1^2) - (2 \times Z1^2 + 2 \times a_j \times X1^2 - Y1^2)^2 $
  3. $ Z3 = (2 \times Z1^2 + 2 \times a_j \times X1^2 - Y1^2)^2 $
  4. $ T3 = (2 \times X1 \times Y1)^2 $

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

More details are in [HKCD09,§3.1].

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