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.^[1] 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:

${\displaystyle Q:\{Q_{1}(X_{0},X_{1},X_{2},X_{3})=0\}\cap \{Q_{2}(X_{0},X_{1},X_{2},X_{3})=0\))$

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:

${\displaystyle \Phi :(x,y)\mapsto (X,Y,Z,T)=(x,y,1,x^{2})}$

We see that the following system of equations holds:

${\displaystyle \mathbf {S} :{\begin{cases}X^{2}-TZ=0\\Y^{2}-aXZ-bZ^{2}-TX=0\end{cases))}$

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

${\displaystyle \mathbf {S} 1:{\begin{cases}X^{2}+Y^{2}-T^{2}=0\\kX^{2}+Z^{2}-T^{2}=0\end{cases))}$.

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

${\displaystyle (x,y)\mapsto (X,Y,Z,T)=(-2y,x^{2}-j,x^{2}+2jx+j,x^{2}+2x+j)}$

${\displaystyle O=(0:1:0)\mapsto (0,1,1,1)}$

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:

${\displaystyle X_{3}=T_{1}Y_{2}X_{1}Z_{2}+Z_{1}X_{2}Y_{1}T_{2))$

${\displaystyle Y_{3}=T_{1}Y_{2}Y_{1}T_{2}-Z_{1}X_{2}X_{1}Z_{2))$

${\displaystyle Z_{3}=T_{1}Z_{1}T_{2}Z_{2}-kX_{1}Y_{1}X_{2}Y_{2))$

${\displaystyle T_{3}=(T_{1}Y_{2})^{2}+(Z_{1}X_{2})^{2))$

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:

${\displaystyle X_{3}=2Y_{1}T_{1}Z_{1}X_{1))$

${\displaystyle Y_{3}=(T_{1}Y_{1})^{2}-(T_{1}Z_{1})^{2}+(Z_{1}Y_{1})^{2))$

${\displaystyle Z_{3}=(T_{1}Z_{1})^{2}-(T_{1}Y_{1})^{2}+(Z_{1}Y_{1})^{2))$

${\displaystyle T_{3}=(T_{1}Z_{1})^{2}+(T_{1}Y_{1})^{2}-(Z_{1}Y_{1})^{2))$

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.^[2] 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^[2]

Example of addition and doubling

Let K = R or C and consider the case:

${\displaystyle \mathbf {S} 1:{\begin{cases}X^{2}+Y^{2}-T^{2}=0\\4X^{2}+Z^{2}-T^{2}=0\end{cases))}$

Consider the points ${\displaystyle P_{1}=(1,{\sqrt {3)),0,2)}$ and ${\displaystyle P_{2}=(1,2,1,{\sqrt {5)))}$: 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:

${\displaystyle X_{3}=T_{1}Y_{2}X_{1}Z_{2}+Z_{1}X_{2}Y_{1}T_{2}=4}$

${\displaystyle Y_{3}=T_{1}Y_{2}Y_{1}T_{2}-Z_{1}X_{2}X_{1}Z_{2}=4{\sqrt {15))}$

${\displaystyle Z_{3}=T_{1}Z_{1}T_{2}Z_{2}-kX_{1}Y_{1}X_{2}Y_{2}=-8{\sqrt {3))}$

${\displaystyle T_{3}=(T_{1}Y_{2})^{2}+(Z_{1}X_{2})^{2}=16}$

The resulting point is ${\displaystyle P_{3}=(4,4{\sqrt {15)),-8{\sqrt {3)),16)}$.

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

${\displaystyle X_{3}=2Y_{1}T_{1}Z_{1}X_{1}=0}$

${\displaystyle Y_{3}=(T_{1}Y_{1})^{2}-(T_{1}Z_{1})^{2}+(Z_{1}Y_{1})^{2}=12}$

${\displaystyle Z_{3}=(T_{1}Z_{1})^{2}-(T_{1}Y_{1})^{2}+(Z_{1}Y_{1})^{2}=-12}$

${\displaystyle T_{3}=(T_{1}Z_{1})^{2}+(T_{1}Y_{1})^{2}-(Z_{1}Y_{1})^{2}=12}$

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:

${\displaystyle x_{3}={\frac {y_{2}x_{1}z_{2}+z_{1}x_{2}y_{1)){(y_{2}^{2}+(z_{1}x_{2})^{2})))}$

${\displaystyle y_{3}={\frac {y_{2}y_{1}-z_{1}x_{2}x_{1}z_{2)){(y_{2}^{2}+(z_{1}x_{2})^{2})))}$

${\displaystyle z_{3}={\frac {z_{1}z_{2}-ax_{1}y_{1}x_{2}y_{2)){(y_{2}^{2}+(z_{1}x_{2})^{2})))}$

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:

${\displaystyle \mathbf {S} 1:{\begin{cases}x^{2}+y^{2}=1\\kx^{2}+z^{2}=1\end{cases))}$

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

${\displaystyle x=X/T}$

${\displaystyle y=Y/T}$

${\displaystyle z=Z/T}$

${\displaystyle XY=X\cdot Y}$

${\displaystyle ZT=Z\cdot T}$

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

${\displaystyle (p,0)\mapsto (0:-1:1)}$

${\displaystyle (x,y)\neq (p,0)\mapsto (2(x-p):(2x+p)(x-p)^{2}-y^{2}:y)}$

${\displaystyle O\mapsto (0:1:1)}$^[3]

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

${\displaystyle C:\ Y^{2}=eX^{4}-2dX^{2}Z^{2}+Z^{4))$^[3]

where:

${\displaystyle e={\frac {-(3p^{2}+4a)}{16)),\ \ d={\frac {3p}{4))}$.

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:

${\displaystyle y^{2}=ex^{4}+2ax^{2}+1}$,

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 ${\displaystyle P_{1}=(x_{1},y_{1})}$ and ${\displaystyle P_{2}=(x_{2},y_{2})}$, their sum is a point ${\displaystyle P_{3}=(x_{3},y_{3})}$, such that:

${\displaystyle x_{3}={\frac {x_{1}y_{2}+y_{1}x_{2)){1-e(x_{1}x_{2})^{2))))$

${\displaystyle y_{3}={\frac {((1+e(x_{1}x_{2})^{2})(y_{1}y_{2}+2ax_{1}x_{2})+2ex_{1}x_{2}({x_{1))^{2}+{x_{2))^{2}))}{(1-e(x_{1}x_{2})^{2})^{2))))$

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:

${\displaystyle X_{3}=X_{1}Z_{1}Y_{2}+Y_{1}X_{2}Z_{2))$

${\displaystyle Y_{3}=\left(Z_{1}^{2}Z_{2}^{2}+eX_{1}^{2}X_{2}^{2}\right)\left(Y_{1}Y_{2}-2aX_{1}X_{2}Z_{1}Z_{2}\right)\ +\ 2eX_{1}X_{2}Z_{1}Z_{2}\left(X_{1}^{2}Z_{2}^{2}+Z_{1}^{2}X_{2}^{2}\right)}$

${\displaystyle Z_{3}=Z_{1}^{2}Z_{2}^{2}-eX_{1}^{2}X_{2}^{2))$

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 ^[4] 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:

${\displaystyle C:\ Y^{2}=X^{4}+Z^{4))$

Choosing two points ${\displaystyle P_{1}=(1:{\sqrt {2)):1)}$ and ${\displaystyle P_{2}=(2:{\sqrt {17)):1)}$, it is possible to find their sum P₃ = P₁ + P₂ using the formulae for adding given above:

${\displaystyle X_{3}=1\cdot 1\cdot {\sqrt {17))+{\sqrt {2))\cdot 2\cdot 1={\sqrt {17))+2{\sqrt {2))}$

${\displaystyle Y_{3}=\left(1^{2}\cdot 1^{2}+1\cdot 1^{2}\cdot 2^{2}\right)\left({\sqrt {2))\cdot {\sqrt {17))-2\cdot 0\cdot 1\cdot 2\cdot 1\cdot 1\right)+2\cdot 1\cdot 1\cdot 2\cdot 1\cdot 1\left(1^{2}\cdot 1^{2}+1^{2}\cdot 2^{2}\right)=5{\sqrt {34))+20}$

${\displaystyle Z_{3}=1^{2}\cdot 1^{2}-1\cdot 1^{2}\cdot 2^{2}=-3}$.

So

${\displaystyle P_{3}=({\sqrt {17))+2{\sqrt {2)):5{\sqrt {34))+20:-3)}$.

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

${\displaystyle X_{3}=1\cdot 1\cdot {\sqrt {2))+{\sqrt {2))\cdot 1\cdot 1=2{\sqrt {2))}$

${\displaystyle Y_{3}=\left(1+1\cdot 1\right)\left({\sqrt {2))\cdot {\sqrt {2))-2\cdot 0\cdot 1\cdot 1\cdot 1\cdot 1\right)+2\cdot 1\left(1^{2}\cdot 1^{2}+1^{2}\cdot 1^{2}\right)=8}$

${\displaystyle Z_{3}=1^{2}\cdot 1^{2}-1\cdot 1^{2}\cdot 1^{2}=0}$

So

${\displaystyle P_{4}=(2{\sqrt {2)):8:0)}$.

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

${\displaystyle y^{2}=x^{4}+2ax^{2}+1}$

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:

${\displaystyle x=X/Z}$

${\displaystyle y=Y/ZZ}$

${\displaystyle XX=X^{2))$

${\displaystyle ZZ=Z^{2))$

${\displaystyle R=2\cdot X\cdot Z}$

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:

${\displaystyle x=X/Z}$

${\displaystyle y=Y/Z^{2))$

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

${\displaystyle x=X/Z}$

${\displaystyle y=Y/ZZ}$

${\displaystyle XX=X^{2))$

${\displaystyle ZZ=Z^{2))$

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

${\displaystyle x=X/Z}$

${\displaystyle y=Y/ZZ}$

${\displaystyle XX=X^{2))$

${\displaystyle ZZ=Z^{2))$

${\displaystyle R=2\cdot X\cdot Z}$

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

${\displaystyle x=X/Z}$

${\displaystyle y=Y/Z^{2))$.