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.

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.