Hessian form of an elliptic curve explained

In geometry, the Hessian curve is a plane curve similar to folium of Descartes. It is named after the German mathematician Otto Hesse.This curve was suggested for application in elliptic curve cryptography, because arithmetic in this curve representation is faster and needs less memory than arithmetic in standard Weierstrass form.^[1]

Definition

Let

K be a field and consider an elliptic curve E in the following special case of Weierstrass form over K :$$Y^2+a\_1\; XY+a\_3\; Y=X^3$$where the curve has discriminant \Delta = \left(a ^{3} _{3} \left(a ^{3} _{1} - 27 a _{3\right)\right)} = a ^{3} _{3} \delta. Then the point P = ( 0 , 0 ) has order 3.

To prove that

P = ( 0 , 0 ) has order 3, note that the tangent to E at P is the line Y =0 which intersects E with multiplicity 3 at P .

Conversely, given a point

P of order 3 on an elliptic curve E both defined over a field K one can put the curve into Weierstrass form with P = ( 0 , 0 ) so that the tangent at P is the line Y = 0 . Then the equation of the curve is Y ^{2+ a} _{1} XY + a _{3} Y = X ^{3} with a _{1,a} _{3\in} K .

To obtain the Hessian curve, it is necessary to do the following transformation:

First let

\mu denote a root of the polynomial$$T^3-\backslash delta\; T^2+T+\; a\_3\backslash delta^2=0.$$

Then$$\backslash mu=.$$

Note that if

K has a finite field of order q \equiv 2 \pmod{ 3 } , then every element of K has a unique cube root; in general, \mu lies in an extension field of K.

Now by defining the following value $D=\backslash frac$ another curve, C, is obtained, that is birationally equivalent to E:

$$C\; :\; x^3\; +\; y^3\; +\; z^3=\; 3Dxyz$$

which is called cubic Hessian form (in projective coordinates)$$C\; :\; x^3\; +\; y^3\; +\; 1=\; 3Dxy$$

in the affine plane (satisfying $x=\backslash frac$ and $y=\backslash frac$).

Furthermore,

D ^{3\ne1} (otherwise, the curve would be singular).

Starting from the Hessian curve, a birationally equivalent Weierstrass equation is given by$$v^2\; =\; u^3\; -\; 27D\backslash left(D^3\; +\; 8\backslash right)u\; +\; 54\backslash left(D^6\; -\; 20\; D^3\; -\; 8\backslash right),$$under the transformations: $$(x,y)\; =\; \backslash left(\backslash eta\; \backslash left(u\; +\; 9D^2\backslash right),\; -\; 1\; +\; \backslash eta\backslash left(3D^3\; -\; Dx\; -12\backslash right)\backslash right)$$and$$(u,v)\; =\; \backslash left(-9D^2\; +\; \backslash varepsilon\; x,\; 3\backslash varepsilon\backslash left(y\; -\; 1\backslash right)\backslash right)$$where: $$\backslash eta\; =\; \backslash frac$$and $$\backslash varepsilon\; =\; \backslash frac$$

Group law

It is interesting to analyze the group law of the elliptic curve, defining the addition and doubling formulas (because the SPA and DPA attacks are based on the running time of these operations). Furthermore, in this case, we only need to use the same procedure to compute the addition, doubling or subtraction of points to get efficient results, as said above. In general, the group law is defined in the following way: if three points lie in the same line then they sum up to zero. So, by this property, the group laws are different for every curve.

In this case, the correct way is to use the Cauchy-Desboves´ formulas, obtaining the point at infinity, that is, the neutral element (the inverse of is again). Let be a point on the curve. The line

y = - x + (x _{1+y} _{1)} contains the point and the point at infinity . Therefore, is the third point of the intersection of this line with the curve. Intersecting the elliptic curve with the line, the following condition is obtained x _{2-(x} _{1+y} _{1) ⋅} x + x _{1 ⋅} y _{1=\theta}

Since

x _{1+y} _{1+D} is non zero (because is distinct to 1), the -coordinate of is and the -coordinate of is, i.e., - P = (y _{1,x} _{1)} or in projective coordinates - P = (Y _{1:X} _{1:Z} _{1)} .

In some application of elliptic curve cryptography and the elliptic curve method of factorization (ECM) it is necessary to compute the scalar multiplications of, say for some integer, and they are based on the double-and-add method; these operations need the addition and doubling formulas.

Doubling

Now, if

P = (X _{1:Y} _{1:Z} _{1)} is a point on the elliptic curve, it is possible to define a "doubling" operation using Cauchy-Desboves´ formulae:

$$[2]P=\backslash left(Y\_1\; \backslash cdot\; \backslash left(X\_1^3-Z\_1^3\backslash right)\; :\; X\_1\; \backslash cdot\; \backslash left(Z\_1^3-Y\_1^3\backslash right)\; :\; Z\_1\; \backslash cdot\; \backslash left(Y\_1^3-X\_1^3\backslash right)\backslash right)$$

Addition

In the same way, for two different points, say

P = (X _{1} : Y _{1} : Z _{1)} and Q = (X _{2} : Y _{2} : Z _{2)} , it is possible to define the addition formula. Let denote the sum of these points,, then its coordinates are given by:

$$R=\backslash left(Y\_1^2\backslash cdot\; X\_2\backslash cdot\; Z\_2-Y\_2^2\backslash cdot\; X\_1\backslash cdot\; Z\_1\; :\; X\_1^2\backslash cdot\; Y\_2\backslash cdot\; Z\_2-X\_2^2\backslash cdot\; Y\_1\backslash cdot\; Z\_1\; :\; Z\_1^2\backslash cdot\; X\_2\backslash cdot\; Y\_2-Z\_2^2\backslash cdot\; X\_1\backslash cdot\; Y\_1\backslash right)$$

Algorithms and examples

There is one algorithm that can be used to add two different points or to double; it is given by Joye and Quisquater. Then, the following result gives the possibility the obtain the doubling operation by the addition:

Proposition. Let be a point on a Hessian elliptic curve . Then:

Furthermore, we have .

Finally, contrary to other parameterizations, there is no subtraction to compute the negation of a point. Hence, this addition algorithm can also be used for subtracting two points and on a Hessian elliptic curve:

To sum up, by adapting the order of the inputs according to equation (2) or (3), the addition algorithm presented above can be used indifferently for:Adding 2 (diff.) points, Doubling a point and Subtracting 2 points with only 12 multiplications and 7 auxiliary variables including the 3 result variables. Before the invention of Edwards curves, these results represent the fastest known method for implementing the elliptic curve scalar multiplication towards resistance against side-channel attacks.

For some algorithms protection against side-channel attacks is not necessary. So, for these doublings can be faster. Since there are many algorithms, only the best for the addition and doubling formulas is given here, with one example for each one:

Addition

Let and be two points distinct to . Assuming that then the algorithm is given by:

The cost needed is 8 multiplications and 3 additions readdition cost of 7 multiplications and 3 additions, depending on the first point.

ExampleGiven the following points in the curve for and, then if we have:

Then:

Doubling

Let be a point, then the doubling formula is given by:

The cost of this algorithm is

ExampleIf

P = ( -1 : -1 : 1 ) is a point over the Hessian curve with parameter, then the coordinates of 2 P = (X:Y:Z) are given by:

That is,

2 P = ( -4 : -2 : 0 )

Extended coordinates

There is another coordinates system with which a Hessian curve can be represented; these new coordinates are called extended coordinates. They can speed up the addition and doubling. To have more information about operations with the extended coordinates see:

http://hyperelliptic.org/EFD/g1p/auto-hessian-extended.html#addition-add-20080225-hwcd

x and y are represented by X , Y , Z , XX , YY , ZZ , XY , YZ , XZ satisfying the following equations:x = X/Z y = Y/Z XX = X ⋅ X YY = Y ⋅ Y ZZ = Z ⋅ Z XY =2 ⋅ X ⋅ Y YZ =2 ⋅ Y ⋅ Z XZ =2 ⋅ X ⋅ Z

See also

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

External links

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

References

• Otto Hesse (1844), "Über die Elimination der Variabeln aus drei algebraischen Gleichungen vom zweiten Grade mit zwei Variabeln", Journal für die reine und angewandte Mathematik, 10, pp. 68 - 96
• Book: Marc Joye, Jean-Jacques Quisquater . Cryptographic Hardware and Embedded Systems — CHES 2001 . 2001 . Hessian Elliptic Curves and Side-Channel Attacks . Lecture Notes in Computer Science . 2162 . 402–410 . Springer-Verlag Berlin Heidelberg 2001 . 10.1007/3-540-44709-1_33 . 978-3-540-42521-2 .
• Book: N. P. Smart . Cryptographic Hardware and Embedded Systems — CHES 2001 . 2001 . The Hessian form of an Elliptic Curve . Lecture Notes in Computer Science . 2162 . 118–125 . Springer-Verlag Berlin Heidelberg 2001 . 10.1007/3-540-44709-1_11 . 978-3-540-42521-2 . 30487134 .

Notes and References

1. Cauchy-Desbove's Formulae: Hessian-elliptic Curves and Side-Channel Attacks, Marc Joye and Jean-Jacques Quisquarter