Jacobian curve

In mathematics, the Jacobi curve is a representation of an elliptic curve different from the usual one defined by the Weierstrass equation. Sometimes it is used in cryptography instead of the Weierstrass form because it can provide a defence against simple and differential power analysis style (SPA) attacks; it is possible, indeed, to use the general addition formula also for doubling a point on an elliptic curve of this form: in this way the two operations become indistinguishable from some side-channel information. The Jacobi curve also offers faster arithmetic compared to the Weierstrass curve.

The Jacobi curve can be of two types: the Jacobi intersection, that is given by an intersection of two surfaces, and the Jacobi quartic.

Elliptic Curves: Basics

Given an elliptic curve, it is possible to do some "operations" between its points: for example one can add two points P and Q obtaining the point P + Q that belongs to the curve; given a point P on the elliptic curve, it is possible to "double" P, that means find [2]P = P + P (the square brackets are used to indicate [n]P, the point P added n times), and also find the negation of P, that means find –P. In this way, the points of an elliptic curve forms a group. Note that the identity element of the group operation is not a point on the affine plane, it only appears in the projective coordinates: then O = (0: 1: 0) is the "point at infinity", that is the neutral element in the group law. Adding and doubling formulas are useful also to compute [n]P, the n-th multiple of a point P on an elliptic curve: this operation is considered the most in elliptic curve cryptography.

An elliptic curve E, over a field K can be put in the Weierstrass form y² = x³ + ax + b, with a, b in K. What will be of importance later are point of order 2, that is P on E such that [2]P = O and P ≠ O. If P = (p, 0) is a point on E, then it has order 2; more generally the points of order 2 correspond to the roots of the polynomial f(x) = x³ + ax + b.

From now on, we will use E_a,b to denote the elliptic curve with Weierstrass form y² = x³ + ax + b.

If E_a,b is such that the cubic polynomial x³ + ax + b has three distinct roots in K and b = 0 we can write E_a,b in the Legendre normal form:

E_a,b: y² = x(x + 1)(x + j)

In this case we have three points of order two: (0, 0), (–1, 0), (–j, 0). In this case we use the notation E[j]. Note that j can be expressed in terms of a, b.

Definition: Jacobi intersection

An elliptic curve in P³(K) can be represented as the intersection of two quadric surfaces:

It is possible to define the Jacobi form of an elliptic curve as the intersection of two quadrics. Let E_a,b be an elliptic curve in the Weierstrass form, we apply the following map to it:

We see that the following system of equations holds:

The curve E[j] corresponds to the following intersection of surfaces in P³(K):

.

The "special case", E[0], the elliptic curve has a double point and thus it is singular.

S1 is obtained by applying to E[j] the transformation:

ψ: E[j] → S1

Group law

For S1, the neutral element of the group is the point (0, 1, 1, 1), that is the image of O = (0: 1: 0) under ψ.

Addition and doubling

Given P₁ = (X₁, Y₁, Z₁, T₁) and P₂ = (X₂, Y₂, Z₂, T₂), two points on S1, the coordinates of the point P₃ = P₁ + P₂ are:

These formulas are also valid for doubling: it sufficies to have P₁ = P₂. So adding or doubling points in S1 are operations that both require 16 multiplications plus one multiplication by a constant (k).

It is also possible to use the following formulas for doubling the point P₁ and find P₃ = [2]P₁:

Using these formulas 8 multiplications are needed to double a point. However, there are even more efficient “strategies” for doubling that require only 7 multiplications. In this way it is possible to triple a point with 23 multiplications; indeed [3]P₁ can be obtained by adding P₁ with [2]P₁ with a cost of 7 multiplications for [2]P₁ and 16 for P₁ + [2]P₁

Example of addition and doubling

Let K = R or C and consider the case:

Consider the points and : it is easy to verify that P₁ and P₂ belong to S1 (it is sufficient to see that these points satisfy both equations of the system S1).

Using the formulas given above for adding two points, the coordinates for P₃, where P₃ = P₁ + P₂ are:

The resulting point is .

With the formulas given above for doubling, it is possible to find the point P₃ = [2]P₁:

So, in this case P₃ = [2]P₁ = (0, 12, –12, 12).

Negation

Given the point P₁ = (X₁, Y₁, Z₁, T₁) in S1, its negation is −P₁ = (−X₁, Y₁, Z₁, T₁)

Addition and doubling in affine coordinates

Given two affine points P₁ = (x₁, y₁, z₁) and P₂ = (x₂, y₂, z₂), their sum is a point P₃ with coordinates:

These formulas are valid also for doubling with the condition P₁ = P₂.

Extended coordinates

There is another kind of coordinate system with which a point in the Jacobi intersection can be represented. Given the following elliptic curve in the Jacobi intersection form:

the extended coordinates describe a point P = (x, y, z) with the variables X, Y, Z, T, XY, ZT, where:

Sometimes these coordinates are used, because they are more convenient (in terms of time-cost) in some specific situations. For more information about the operations based on the use of these coordinates see http://hyperelliptic.org/EFD/g1p/auto-jintersect-extended.html

Definition: Jacobi quartic

An elliptic curve in Jacobi quartic form can be obtained from the curve E_a,b in the Weierstrass form with at least one point of order 2. The following transformation f sends each point of E_a,b to a point in the Jacobi coordinates, where (X: Y: Z) = (sX: s²Y: sZ).

f: E_a,b → J

Applying f to E_a,b, one obtains a curve in J of the following form:

where:

.

are elements in K. C represents an elliptic curve in the Jacobi quartic form, in Jacobi coordinates.

Jacobi quartic in affine coordinates

The general form of a Jacobi quartic curve in affine coordinates is:

,

where often e = 1 is assumed.

Group law

The neutral element of the group law of C is the projective point (0: 1: 1).

Addition and doubling in affine coordinates

Given two affine points and , their sum is a point , such that:

As in the Jacobi intersections, also in this case it is possible to use this formula for doubling as well.

Addition and doubling in projective coordinates

Given two points P₁ = (X₁: Y₁: Z₁) and P₂ = (X₂: Y₂: Z₂) in C′, the coordinates for the point P₃ = (X₃: Y₃: Z₃), where P₃ = P₁ + P₂, are given in terms of P₁ and P₂ by the formulae:

One can use this formula also for doubling, with the condition that P₂ = P₁: in this way the point P₃ = P₁ + P₁ = [2]P₁ is obtained.

The number of multiplications required to add two points is 13 plus 3 multiplications by constants: in particular there are two multiplications by the constant e and one by the constant a.

There are some "strategies" to reduce the operations required for adding and doubling points: the number of multiplications can be decreased to 11 plus 3 multiplications by constants (see section 3 for more details).

The number of multiplications can be reduced by working on the constants e and d: the elliptic curve in the Jacobi form can be modified in order to have a smaller number of operations for adding and doubling. So, for example, if the constant d in C is significantly small, the multiplication by d can be cancelled; however the best option is to reduce e: if it is small, not only one, but two multiplications are neglected.

Example of addition and doubling

Consider the elliptic curve E_4,0, it has a point P of order 2: P = (p, 0) = (0, 0). Therefore, a = 4, b = p = 0 so we have e = 1 and d = 1 and the associated Jacobi quartic form is:

Choosing two points and , it is possible to find their sum P₃ = P₁ + P₂ using the formulae for adding given above:

.

So

.

Using the same formulae, the point P₄ = [2]P₁ is obtained:

So

.

Negation

The negation of a point P₁ = (X₁: Y₁: Z₁) is: −P₁ = (−X₁: Y₁: Z₁)

Alternative coordinates for the Jacobi quartic

There are other systems of coordinates that can be used to represent a point in a Jacobi quartic: they are used to obtain fast computations in certain cases. For more information about the time-cost required in the operations with these coordinates see http://hyperelliptic.org/EFD/g1p/auto-jquartic.html

Given an affine Jacobi quartic

the Doubling-oriented XXYZZ coordinates introduce an additional curve parameter c satisfying a² + c² = 1 and they represent a point (x, y) as (X, XX, Y, Z, ZZ, R), such that:

the Doubling-oriented XYZ coordinates, with the same additional assumption (a² + c² = 1), represent a point (x, y) with (X, Y, Z) satisfying the following equations:

Using the XXYZZ coordinates there is no additional assumption, and they represent a point (x, y) as (X, XX, Y, Z, ZZ) such that:

while the XXYZZR coordinates represent (x, y) as (X, XX, Y, Z, ZZ, R) such that:

with the XYZ coordinates the point (x, y) is given by (X, Y, Z), with:

.

See also

For more information about the running-time required in a specific case, see Table of costs of operations in elliptic curves.

Notes

References

• http://hyperelliptic.org/EFD/index.html

External links

• http://hyperelliptic.org/EFD/g1p/index.html