Twisted Edwards curve

A twisted Edwards curve E_{E,a,d} over a field \mathbb{K} with characteristic not equal to 2 (that is, no element is its own additive inverse) is an affine plane curve defined by the equation: : E_{E,a,d}: a x^2+y^2= 1+dx^2y^2 where a, d are distinct non-zero elements of \mathbb{K}. Each twisted Edwards curve is a twist of an Edwards curve. The special case a = 1 is untwisted, because the curve reduces to an ordinary Edwards curve. 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 \mathbb{K} be a field with characteristic different from 2. Let (x_1,y_1) and (x_2,y_2) be points on the twisted Edwards curve. The equation of twisted Edwards curve is written as; : E_{E, a, d}: ax^2+y^2=1+dx^2y^2. The sum of these points (x_1,y_1), (x_2,y_2) on E_{E, a, d} is: : (x_1,y_1) + (x_2,y_2) = \left(\frac{x_1y_2+y_1x_2}{1+dx_1x_2y_1y_2} , \frac{y_1y_2-ax_1x_2}{1-dx_1x_2y_1y_2}\right) The neutral element is (0,1) and the negative of (x_1,y_1) is (-x_1,y_1) These formulas also work for doubling. If a is a square in \mathbb{K} and d is a non-square in \mathbb{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: ; 3x^2 + y^2 = 1 + 2x^2y^2 it is possible to add the points P_1=(1,\sqrt{2}) and P_2=(1,-\sqrt{2}) using the formula given above. The result is a point P3 that has coordinates: : x_3 = \frac{x_1y_2+y_1x_2}{1+dx_1x_2y_1y_2} = 0, : y_3 = \frac{y_1y_2-ax_1x_2}{1-dx_1x_2y_1y_2} = -1. Doubling on twisted Edwards curves Doubling can be performed with exactly the same formula as addition. Doubling of a point (x_1,y_1) on the curve E_{E, a, d} is: 2(x_1,y_1) = (x_3,y_3) where : \begin{align} x_3 & = \frac{x_1y_1+y_1x_1}{1+dx_1x_1y_1y_1}=\frac{2x_1y_1}{ax_1^2+y_1^2} \\[6pt] y_3 & = \frac{y_1y_1-ax_1x_1}{1-dx_1x_1y_1y_1}=\frac{y_1^2-ax_1^2}{2-ax_1^2-y_1^2}. \end{align} Denominators in doubling are simplified using the curve equation dx^2y^2 = ax^2 + by^2 - 1. This reduces the power from 4 to 2 and allows for more efficient computation. 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 P_1=(1,\sqrt{2}). The point 2P1 obtained using the formula above has the following coordinates: : x_3 = \frac{2x_1y_1}{ax_1^2+y_1^2} = \frac{2\sqrt{2}}{5}, : y_3 = \frac{y_1^2-ax_1^2}{2-ax_1^2-y_1^2} = \frac{1}{3}. It is easy to see, with some little computations, that the point P_3=\left(\frac{2\sqrt{2}}{5}, \frac{1}{3}\right) belongs to the curve 3x^2 + y^2 = 1 + 2x^2y^2. ==Extended coordinates==

There is another kind of coordinate system with which a point in the twisted Edwards curves can be represented. A point (x,y,z) on ax^2+y^2= 1+dx^2y^2 is represented as X, Y, Z, T satisfying the following equations x = X/Z, y = Y/Z, xy = T/Z. 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 (X_1:Y_1:Z_1) are called the inverted twisted Edwards coordinates on the curve (X^2+aY^2)Z^2= X^2Y^2+dZ^4with X_1Y_1Z_1 \ne 0; this point to the affine one (Z_1/X_1, Z_1/Y_1) on E_{E, a, d}. Bernstein and Lange introduced these inverted coordinates, for the case a=1 and observed that the coordinates save time in addition. Projective twisted Edwards coordinates The equation for the projective twisted Edwards curve is given as: (aX^2+Y^2)Z^2= Z^4+dX^2Y^2 For Z1 ≠ 0 the point (X1:Y1:Z1) represents the affine point (x1 = X1/Z1, y1 = Y1/Z1) on EE,a,d. Expressing an elliptic curve in twisted Edwards form saves time in arithmetic, even when the same curve can be expressed in the Edwards form. ==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. ;Algorithm :A = Z1 · Z2, :B = A2 :C = X1 · X2 :D = Y1 · Y2 :E = dC · D :F = B − E :G = B + E :X3 = A · F((X1 + Y1) · (X2 + Y2) − C − D) :Y3 = A · G · (D − aC) :Z3 = F · G 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. ;Algorithm: :B = (X1 + Y1)2 :C = X12 :D = Y12 :E = aC :F = E + D :H = Z12 :J = F − 2H :X3 = (B − C − D).J :Y3 = F · (E − D) :Z3 = F · J ==See also==