Tripling-oriented Doche–Icart–Kohel curve

The tripling-oriented Doche–Icart–Kohel curve is a form of an elliptic curve that has been used lately in cryptography[when?

]; it is a particular type of Weierstrass curve.

At certain conditions some operations, as adding, doubling or tripling points, are faster to compute using this form.

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

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: with

The "point at infinity" represents the neutral element for the group law and it is written in projective coordinates as O = (0:1:0).

The negation of a point P = (x, y) with respect to this neutral element is −P = (x, −y).

Consider an elliptic curve in the Tripling-oriented Doche-Icart-Kohel form in affine coordinates: As in other forms of elliptic curves, it is possible to define some "operations" between points, such as adding points, or doubling (See also The group law).

In the following sections formulas to add, negate and doubling points are given.

The addition and doubling formulas are often used for other operations: given a point P on an elliptic curve it is possible to compute [n]P, where n is an integer, using addition and doubling; computing multiples of points is important in elliptic curve cryptography and in Lenstra elliptic curve factorization.

, its negation with respect to the neutral element

There are also other formulas given in [2] for Tripling-oriented Doche–Icart–Kohel curves for fast tripling operation and mixed-addition.

For computing on these curves points are usually represented in new Jacobian coordinates (Jn): a point in the new Jacobian coordinates is of the form

This means, for example, that the point

is written in the new Jacobian coordinates as

of a point on the curve makes the mixed addition (that is the addition between two points in different systems of coordinates) more efficient.

The neutral element in new Jacobian coordinates is

The following algorithm represents the sum of two points

on an elliptic curve in the Tripling-oriented Doche-Icart-Kohel form.

The cost of this implementation is 7M + 4S + 1*a3 + 10add + 3*2 + 1*4, where M indicates the multiplications, S the squarings, a3 indicates the multiplication by the constant a3, add represents the number of additions required.

affine points on the elliptic curve over

The following algorithm represents the doubling of a point

on an elliptic curve in the Tripling-oriented Doche-Icart-Kohel form.

The cost of this implementation is 2M + 7S + 1*a2 + 1*a3 + 12add + 2*2 + 1*3 + 1*8; here M represents the multiplications, S the squarings, a2 and a3 indicates the multiplications by the constants a2 and a3 respectively, and add indicates the additions.

Notice that here the point is in affine coordinates, so

Any elliptic curve is birationally equivalent to another written in the Weierstrass form.

The following twisted tripling-oriented Doche-Icart-Kohel curve: can be transformed into the Weierstrass form by the map: In this way

becomes: Conversely, given an elliptic curve in the Weierstrass form: it is possible to find the "corresponding" Tripling-oriented Doche–Icart–Kohel curve, using the following relation: where a is a root of the polynomial where is the j-invariant of the elliptic curve

Notice that in this case the map given is not only a birational equivalence, but an isomorphism between curves.