Jacobian curve - AbsoluteAstronomy.com

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Jacobi curve is a representantion of an elliptic curve

Elliptic curve

In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...

different than the usual one (Weierstrass equation

Elliptic curve

). Sometimes it is used in cryptography

Cryptography

Cryptography is the practice and study of techniques for secure communication in the presence of third parties...

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 offers also 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.

Definition: Jacobi intersection

Let

be a field

Field (mathematics)

In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

.
An elliptic curve in the projective space

Projective space

In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....

over

can be represented as the intersection

Intersection

Intersection has various meanings in different contexts:*In mathematics and geometry**Intersection , the set of elements common to some collection of sets.**Line-line intersection**Line-plane intersection**Line–sphere intersection...

of two quadric surfaces

Quadric

In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface in -dimensional space defined as the locus of zeros of a quadratic polynomial...

It is possible to define the Jacobi form of an elliptic curve as the intersection of two quadrics, using the following map

Map (mathematics)

In most of mathematics and in some related technical fields, the term mapping, usually shortened to map, is either a synonym for function, or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function.In graph theory, a map is a...

applied to the usual Weierstrass curve

This map sends the point

to the point

that satisfies the following system of equations

Simultaneous equations

In mathematics, simultaneous equations are a set of equations containing multiple variables. This set is often referred to as a system of equations. A solution to a system of equations is a particular specification of the values of all variables that simultaneously satisfies all of the equations...

The map

can be applied to an elliptic curve in the Weierstrass form

Elliptic curve

with three points

Point (geometry)

In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...

of order

Order (mathematics)

Order in mathematics may refer to:-In algebra:*Order , the cardinality of a group or period of an element*Order, or degree of a polynomial*Order, or dimension of a matrix*Order , an algebraic structure*Ordered group...

two: this means that the polynomial

Polynomial

In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

has three roots

Equation solving

In mathematics, to solve an equation is to find what values fulfill a condition stated in the form of an equation . These expressions contain one or more unknowns, which are free variables for which values are sought that cause the condition to be fulfilled...

in the field

. Suppose that the roots are

, with

, then the three points of order two are: (0,0), (-1,0), (-j,0) and the curve

can, thus, be written as:

where

is the "point at infinity", that is the neutral element

Identity element

In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...

in the group law.

The curve

corresponds to the following intersection of surface

Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...

s in

.

The "special case" is when j=0: the elliptic curve has a double point and thus it is singular

Singular point of an algebraic variety

In mathematics, a singular point of an algebraic variety V is a point P that is 'special' , in the geometric sense that V is not locally flat there. In the case of an algebraic curve, a plane curve that has a double point, such as the cubic curveexhibits at , cannot simply be parametrized near the...

.

S1 is obtained by applying to

the transformation

Map (mathematics)

Group law

(See also The group law).

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

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

.
Adding and doubling formulas are useful also to compute [n]P, the nth multiple of a point P on an elliptic curve: this operation is considered the most in elliptic curve cryptography

Elliptic curve cryptography

Elliptic curve cryptography is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. The use of elliptic curves in cryptography was suggested independently by Neal Koblitz and Victor S...

.

For S1, the neutral element

Identity element

In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...

of the group is the point

, that is the image of

Addition and doubling

Given

and

, two points on

, the coordinates

Coordinate system

In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...

of the point

are:

These formulas are also valid for doubling: it sufficies to have

. So adding or doubling points in

are operations that both require 16 multiplications plus one multiplication by a constant (

).

It is also possible to use the following formulas for doubling the point

and find

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

can be obtained by adding

with

with a cost of 7 multiplications for

and 16 for

Example of addition and doubling

Let the field

, or

.
Consider the case:

Consider the points

and

: it is easy to verify that

and

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

, where

are:

The resulting point is

.

With the formulas given above for doubling, it is possible to find the point

So, in this case

Negation

Given the point

, its negation

Additive inverse

In mathematics, the additive inverse, or opposite, of a number a is the number that, when added to a, yields zero.The additive inverse of a is denoted −a....

Addition and doubling in affine coordinates

Given two affine points

and

, their sum is a point

with coordinates:

These formulas are valid also for doubling with the condition

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

with the variables

, where:

Sometimes these coordinates are used, because they are more convenient (in terms of time-cost) in some specific situations.

To have 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

Let

be a field

Field (mathematics)

.
An elliptic curve in Jacobi quartic form can be obtained from a curve

in the Weierstrass form with at least one point of order 2:

with

.

Let

be a root of

, then P is a point of order 2 of the elliptic curve; let

be the point at infinity.

The following transformation

Map (mathematics)

sends each point of

to a point in the Jacobi coordinates (

)

Applying

, one obtains a curve in

of the following form:

where

.

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

is assumed.

Group law

The neutral element of the group law of

is the projective point

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

and

, the coordinates for the point

, where

, are given in terms of

and

by the formulae:

One can use this formula also for doubling, with the condition that

: in this way the point

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

and one by the constant

.

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

and

: 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

in C’ is significantly small, the multiplication by

can be cancelled; however the best option is to reduce

: if it is small, not only one, but two multiplications are neglected.

Example of addition and doubling

Consider an elliptic curve in the Weierstrass form in affine coordinates

Affine space

In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...

over the field

, or

, with

and

has a point

of order 2:

.

Applying the function

, the following elliptic curve in projective Jacobi quartic form is obtained:

where

and

.

Choosing two points

and

, it is possible to find their sum

using the formulae for adding given above:

.

So

.

Using the same formulae, the point

is obtained:

so,

Negation

The negation of a point

is:

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