Tripling-oriented Doche–Icart–Kohel curve

The Tripling oriented Doche-Icart-Kohel curve is a particular form of an elliptic curve that has been used lately in cryptography; in some cases, indeed, it is more convenient to use different representations for an elliptic curve, than the usual Weierstrass form: at certain conditions some operations, as adding, doubling or tripling points, require less time-cost.

The Tripling oriented Doche-Icart-Kohel curve, often called with the abbreviation 3DIK has been introduced by Christophe Doche, Thomas Icart, and David R. Kohel in

Definition

Let $K$ be a field of characteristic different form 2 and 3.

An elliptic curve in Tripling oriented Doche-Icart-Kohel form is defined by the equation:

$T_{a}$: $y^{2}=x^{3}+3a(x+1)^{2}$

with Failed to parse (unknown function "\inK"): {\displaystyle a\inK} .

A general point P on $T_{a}$ has affine coordinates $(x,y)$.

Usually this curve is represented with a different coordinates, called the new Jacobian coordinates