Formulas

We can derive many formulas working in several coordinate systems. We give in the following pages formulas for:

Proofs for these formulas, as well as extra structures and other mathematical considerations, are explained at length in the whitepaper. The $$(e,u)$$ coordinates relate to the view of the curve as a Jacobi quartic, as explained in the relevant paper.

In all these formulas, the following apply:

• The base curve $$E$$ has equation $$y^2 = x(x^2 + ax + b)$$. Group $$\mathbb{G}$$ consists of the point of $$E$$ which are not points of $$r$$-torsion.

• The neutral is $$N = (0, 0)$$. The group law is called "addition" and denoted as such. Group $$\mathbb{G}$$ is homomorphic to the subgroup of points of $$r$$-torsion $$E[r]$$ and thus has the same resistance to discrete logarithm.

• In $$(e, u)$$ coordinates, the equation is $$e^2 = (a^2-4b)u^4 - 2au^2 + 1$$. All curve points can now be used (not just $$r$$-torsion points). The neutral element of the group is represented by either $$N = (-1,0)$$ or $$\mathbb{O} = (1,0)$$ (the "point-at-infinity", which is not actually "at infinity" in this coordinate system).