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In algebraic geometry, the twisted Edwards curves are plane models of elliptic curves, a generalisation of Edwards curves introduced by Daniel J. Bernstein et al. in 2007. The curve set is named after mathematician Harold M. Edwards. Elliptic curves are important in public key cryptography and twisted Edwards curves are at the heart of an electronic signature scheme called EdDSA that offers high performance while avoiding security problems that have surfaced in other digital signature schemes.
Contents
- Group law
- Addition on twisted Edwards curves
- Doubling on twisted Edwards curves
- Extended coordinates
- Inverted twisted Edwards coordinates
- Projective twisted Edwards coordinates
- Addition in projective twisted curves
- Doubling on projective twisted curves
- References
As the name suggests, each twisted Edwards curve, is a twist of an Edwards curve. A twisted Edwards curve E E,a,d over a field K which have char(K)≠2 is an affine plane curve defined by the equation:
E E,a,d:where a, d are distinct non-zero elements of K. An Edwards curve is a twisted Edwards curve with a = 1.
Every twisted Edwards curve is birationally equivalent to an elliptic curve in Montgomery form and vice versa.
Group law
As for all elliptic curves, also for the twisted Edwards curve, it is possible to do some operations between its points, such as adding two of them or doubling (or tripling) one. The results of these operations are always points that belong to the curve itself. In the following sections some formulas are given to obtain the coordinates of a point resulted from an addition between two other points (addition), or the coordinates of point resulted from a doubling of a single point on a curve.
Addition on twisted Edwards curves
Let K be a field with characteristic different from 2. Let
EE,a,d:
The sum of these points
The neutral element is (0,1) and the negative of
These formulas also work for doubling. If a is a square in K and d is a non-square in K, these formulas are complete: this means that they can be used for all pairs of points without exceptions; so they work for doubling as well, and neutral elements and negatives are accepted as inputs.
Example of addition
Given the following twisted Edwards curve with a=3 and d=2:
it is possible to add the points
Doubling on twisted Edwards curves
Doubling can be performed with exactly the same formula as addition. Doubling of a point (x1,y1) on the curve EE,a,d is:
[2](x1,y1) = (x3,y3)
where
Example of doubling
Considering the same twisted Edwards curve given in the previous example, with a=3 and d=2, it is possible to double the point
It is easy to see, with some little computations, that the point
Extended coordinates
There is another kind of coordinate system with which a point in the twisted Edwards curves can be represented. A point
The coordinates of the point (X:Y:Z:T) are called the extended twisted Edwards coordinates. The identity element is represented by (0:1:1:0). The negative of a point is (-X:Y:Z:-T).
Inverted twisted Edwards coordinates
The coordinates of the point
Projective twisted Edwards coordinates
The equation for the projective twisted Edwards curve is given as:
Expressing an elliptic curve in twisted Edwards form saves time in arithmetic, even when the same curve can be expressed in the Edwards form. To know more about the speeds of addition and doubling in projective coordinates on Edwards curves, standard coordinates on twisted Edwards curves, inverted coordinates on Edwards curves and inverted coordinates on twisted Edwards curves refer to the table in:
http://hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html
Addition in projective twisted curves
The addition on a projective twisted Edwards curve is given by
(X3:Y3:Z3) = (X1:Y1:Z1)+(X2:Y2:Z2)and costs 10Multiplications + 1Squaring + 2D + 7 additions, where the 2D are one multiplication by a and one by d.
Doubling on projective twisted curves
Doubling on the projective twisted curve is given by
(X3:Y3:Z3) = 2(X1:Y1:Z1).This costs 3Multiplications + 4Squarings + 1D + 7additions, where 1D is a multiplication by a.