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| <!-- Deleted image removed: [[Image:Edwards curve.jpg|300px|right|thumb|An Edward curve of equation <math>x^2+y^2=1-300x^2y^2</math>]] -->
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| In [[mathematics]], an '''Edwards curve''' is a new representation of [[elliptic curve]]s, discovered by [[Harold Edwards (mathematician)|Harold Edwards]] in 2007. The concept of elliptic curves over [[finite fields]] is widely used in [[elliptic curve cryptography]]. Applications of Edwards curves to [[cryptography]] were developed by [[Daniel J. Bernstein|Bernstein]] and Lange: they pointed out several advantages of the Edwards form in comparison to the more well known [[elliptic curve|Weierstrass form]].
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| ==Definition==
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| [[File:Edward-curves.svg|right|thumb|Edwards curves of equation ''x''<sup>2</sup> + ''y''<sup>2</sup> = 1 − ''d'' ·''x''<sup>2</sup>·''y''<sup>2</sup> over the real numbers for ''d'' = 300 (red), ''d'' = √8 (yellow) and ''d'' = −0.9 (blue)]]
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| The equation of an Edwards curve over a [[field (mathematics)|field]] ''K'' which does not
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| have characteristic 2 is:
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| : <math> x^2 + y^2 = 1 + d x^2 y^2 \, </math>
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| for some [[scalar field|scalar]] <math>d\in K\setminus\{0,1\}</math>.
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| Also the following form with parameters ''c'' and ''d'' is called an Edwards curve:
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| : <math> x^2 + y^2 = c^2(1 + dx^2 y^2) \, </math>
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| where ''c'', ''d'' ∈ ''K'' with ''cd''(1 − ''c''<sup>4</sup>·''d'') ≠ 0.
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| Every Edwards curve is [[birationally equivalent]] to an elliptic curve in [[elliptic curve|Weierstrass form]]. If ''K'' is finite, then a sizeable fraction of all elliptic curves over ''K'' can be written as Edwards curves.
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| Often elliptic curves in Edwards form are defined having c=1, without loss of generality. In the following sections, it is assumed that c=1.
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| ==The group law==
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| (See also [[elliptic curve#The group law|Weierstrass curve group law]])
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| It is possible to do some operations on the points on any elliptic curve, such as adding two or more points and doubling or tripling one. Usually, given two points ''P'' and ''Q'' on an elliptic curve, the point ''P'' + ''Q'' is directly related to third point of intersection between the curve and the line that passes trough ''P'' and ''Q''; but in the case of Edwards curve this is not true: indeed the curve expressed in Edwards form has degree 4, so drawing a line one gets not 3 but 4 intersection points. For this case a geometric explanation of the addition law is given in
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| <ref>{{Cite web|url=http://eprint.iacr.org/2009/155|title=Faster Computation of the Tate Pairing|author=Christophe Arene|coauthors=and Tanja Lange and Michael Naehrig and Christophe Ritzenthaler|year=2009|accessdate=28 February 2010}}</ref> | |
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| ===Edwards addition law===
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| [[File:Edwards Curve,Clock group diagram.jpg|300px|right|thumb|Clock group]]
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| It is possible to add points on an elliptic curve, and, in this way, obtain another point that belongs to the curve as well.
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| To understand better the concept of "addition" between points on a curve, a nice example is given by the circle:
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| take the circle of radius 1
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| :<math>{\displaystyle}x^2+y^2=1</math>
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| and consider two points P<sub>1</sub>=(x<sub>1</sub>,y<sub>1</sub>), P<sub>2</sub>=(x<sub>2</sub>,y<sub>2</sub>) on it. Let α<sub>1</sub> and α<sub>2</sub> be the angles such that:
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| :<math>{\displaystyle}P_{1}=(x_{1},y_{1})=(\sin{\alpha_{1}},\cos{\alpha_{1}})</math>
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| :<math>{\displaystyle}P_{2}=(x_{2},y_{2})=(\sin{\alpha_{2}},\cos{\alpha_{2}})</math>
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| The sum of P<sub>1</sub> and P<sub>2</sub> is, thus, given by the sum of "their angles". That is, the point P<sub>3</sub>=P<sub>1</sub>+P<sub>2</sub> is a point on the circle with coordinates (x<sub>3</sub>,y<sub>3</sub>), where:
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| :<math>{\displaystyle}x_{3}=\sin({\alpha}_{1}+{\alpha}_{2})=\sin{\alpha}_{1}\cos{\alpha}_{2}+\sin{\alpha}_{2}\cos{\alpha}_{1}=x_{1}y_{2}+x_{2}y_{1}</math>
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| :<math>{\displaystyle}y_{3}=\cos({\alpha}_{1}+{\alpha}_{2})=\cos{\alpha}_{1}\cos{\alpha}_{2}-\sin{\alpha}_{1}\sin{\alpha}_{2}=y_{1}y_{2}-x_{1}x_{2}.</math>
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| In this way, the addition formula for points on the circle of radius 1 is:
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| :<math>{\displaystyle}(x_1,y_1)+(x_2,y_2) = (x_1y_2+x_2y_1,y_1y_2-x_1x_2)</math>.
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| When two points (''x''<sub>1</sub>, ''y''<sub>1</sub>) and (''x''<sub>2</sub>, ''y''<sub>2</sub>) on an Edwards curve are added, the result is another point which has coordinates:
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| : <math> (x_1,y_1) + (x_2,y_2) = \left( \frac{x_1 y_2 + y_1 x_2}{1 + dx_1 x_2 y_1 y_2}, \frac{y_1 y_2 - x_1 x_2}{1 - dx_1 x_2 y_1 y_2} \right) \, </math>
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| The [[identity element|neutral element]] of this addition is (0, 1). The inverse of any point (''x''<sub>1</sub>, ''y''<sub>1</sub>) is (−''x''<sub>1</sub>, ''y''<sub>1</sub>). The point (0, −1) has order 2: this means that the sum of this point to itself gives the "zero element" that is the [[neutral element]] of the group law, i.e. 2(0, −1) = (0, 1).
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| If ''d is not a square'' in ''K'', then there are no exceptional points: the denominators 1 + ''dx''<sub>1</sub>''x''<sub>2</sub>''y''<sub>1</sub>''y''<sub>2</sub> and 1 − ''dx''<sub>1</sub>''x''<sub>2</sub>''y''<sub>1</sub>''y''<sub>2</sub> are always nonzero. Therefore, the Edwards addition law is complete when ''d'' is not a square in ''K''. This means that the formulas work for all pairs of input points on the edward curve with no exceptions for doubling, no exception for the neutral element, no exception for negatives, etc.<ref name = "Daniel. J Bernstein , Tanja Lange">Daniel. J. Bernstein , Tanja Lange, pag. 3, '' Faster addition and doubling on elliptic curves''</ref> In other words, it is defined for all pairs of input points on the Edwards curve over ''K'' and the result gives the sum of the input points.
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| If ''d is a square'' in ''K'', then the same operation can have exceptional points, i.e. there can be pairs (''x''<sub>1</sub>, ''y''<sub>1</sub>) and (''x''<sub>2</sub>, ''y''<sub>2</sub>) where 1 + ''dx''<sub>1</sub>''x''<sub>2</sub>''y''<sub>1</sub>''y''<sub>2</sub> = 0 or 1 − ''dx''<sub>1</sub>''x''<sub>2</sub>''y''<sub>1</sub>''y''<sub>2</sub> = 0.
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| One of the attractive feature of the Edwards Addition law is that it is strongly ''unified'' i.e. it can also be used to double a point, simplifying protection against [[side-channel attack]]. The addition formula above is faster than other unified formulas and has the strong property of completeness<ref name = "Daniel. J Bernstein , Tanja Lange"/>
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| '''Example of addition law ''':
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| Let's consider the elliptic curve in the Edwards form with ''d''=2
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| :<math>{\displaystyle}x^2 + y^2 = 1 + 2 x^2 y^2</math>
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| and the point <math>P_1=(1,0)</math> on it. It is possible to prove that the sum of ''P''<sub>1</sub> with the neutral element (0,1) gives again P<sub>1</sub>. Indeed, using the formula given above, the coordinates of the point given by this sum are:
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| :<math>x_3 = \frac{x_1 y_2 + y_1 x_2}{1 + dx_1 x_2 y_1 y_2} = 1</math>
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| :<math>y_3 = \frac{y_1 y_2 - x_1 x_2}{1 - dx_1 x_2 y_1 y_2} = 0</math>
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| ==Projective homogeneous coordinates==
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| In the context of cryptography, [[homogeneous coordinates]] are used to prevent [[elliptic curve cryptography|field inversions]] that appear in the affine formula. To avoid inversions in the original Edwards addition formulas, the curve equation can be written in [[projective space|projective coordinates]] as: | |
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| <math>(X^2+Y^2)Z^2=Z^4+dX^2Y^2</math>.
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| A projective point (''X''<sub>1</sub> : ''Y''<sub>1</sub> : ''Z''<sub>1</sub>) corresponds to the [[Affine space|affine point]] (''X''<sub>1</sub>/''Z''<sub>1</sub>, ''Y''<sub>1</sub>/''Z''<sub>1</sub>) on the Edwards curve.
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| The identity element is represented by (0 : 1 : 1). The inverse of (''X''<sub>1</sub> : ''Y''<sub>1</sub> : ''Z''<sub>1</sub>) is (−''X''<sub>1</sub> : ''Y''<sub>1</sub> : ''Z''<sub>1</sub>).
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| An addition formula in projective homogeneous coordinates is given by:
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| : (''X''<sub>3</sub> : ''Y''<sub>3</sub> : ''Z''<sub>3</sub>) = (''X''<sub>1</sub> : ''Y''<sub>1</sub> : ''Z''<sub>1</sub>) + (''X''<sub>2</sub> : ''Y''<sub>2</sub> : ''Z''<sub>2</sub>)
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| where
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| : ''X''<sub>3</sub> = ''Z''<sub>1</sub>''Z''<sub>2</sub>(''X''<sub>1</sub>''Y''<sub>1</sub> − ''Y''<sub>1</sub>''X''<sub>2</sub>)(''X''<sub>1</sub>''Y''<sub>1</sub>''Z''<sub>2</sub><sup>2</sup></sup>2</sub> + ''Z''<sub>1</sub><sup>2</sup>''X''<sub>2</sub>''Y''<sub>2</sub>)
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| : Y<sub>3</sub> = Z<sub>1</sub>Z<sub>2</sub>(X<sub>1</sub>X<sub>2</sub> + Y<sub>1</sub>Y<sub>2</sub>)(X<sub>1</sub>Y<sub>1</sub>Z<sub>2</sub><sup>2</sup> − Z<sub>1</sub><sup>2</sup>X<sub>2</sub>Y<sub>2</sub>)
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| : Z<sub>3</sub> = kZ<sub>1</sub><sup>2</sup>Z<sub>2</sub><sup>2</sup>(X<sub>1</sub>X<sub>2</sub> + Y<sub>1</sub>Y<sub>2</sub>)(X<sub>1</sub>Y<sub>2</sub> − Y<sub>1</sub>X<sub>2</sub>) with ''k'' = 1/''c''.
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| ===Algorithm===
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| Using the following algorithm, X<sub>3</sub>, Y<sub>3</sub>, Z<sub>3</sub> can be written as:
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| X<sub>3</sub>→ GJ , Y<sub>3</sub>→ HK, Z<sub>3</sub>→ kJK.d
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| where:
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| A→ X<sub>1</sub>Z<sub>2</sub>,
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| B→ Y<sub>1</sub>Z<sub>2</sub>,
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| C→ Z<sub>1</sub>X<sub>2</sub>,
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| D→ Z<sub>1</sub>Y<sub>2</sub>,
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| E→ AB,
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| F→ CD,
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| G→ E+F,
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| H→ E-F,
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| J→ (A-C)(B+D)-H,
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| K→ (A+D)(B+C)-G
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| ==Doubling==
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| ''Doubling'' can be performed with exactly the same formula as addition. Doubling refers to the case in which the inputs (''x''<sub>1</sub>, ''y''<sub>1</sub>) and (''x''<sub>2</sub>, ''y''<sub>2</sub>) are known to be equal. Since (''x''<sub>1</sub>, ''y''<sub>1</sub>) is on the Edwards curve, one can substitute the coefficient by (''x''<sub>1</sub><sup>2</sup> + ''y''<sub>1</sub><sup>2</sup> − 1)/''x''<sub>1</sub><sup>2</sup>''y''<sub>1</sub><sup>2</sup> as follows:
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| : <math>
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| \begin{align}
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| 2(x_1,y_1) & = (x_1,y_1)+(x_1,y_1) \\[6pt]
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| 2(x_1,y_1) & = \left( \frac{2x_1y_1}{1+dx_1^2y_1^2}, \frac{y_1^2-x_1^2}{1-dx_1^2y_1^2} \right) \\[6pt]
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| & = \left( \frac{2x_1y_1}{x_1^2+y_1^2}, \frac{y_1^2-x_1^2}{2-x_1^2-y_1^2} \right)
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| \end{align}
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| </math>
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| This reduces the degree of the denominator from 4 to 2 which is reflected in faster doublings.
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| A general addition in Edwards coordinates takes 10'''M''' + 1'''S''' + 1'''C''' + 1'''D''' + 7'''a''' and doubling costs 3'''M''' + 4'''S''' + 3'''C''' + 6'''a''' where '''M''' is field multiplications, '''S''' is field squarings, '''D''' is the cost of multiplying by a selectable curve parameter and '''a''' is field addition.
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| ;Example of doubling
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| As in the previous example for the addition law, let's consider the Edwards curve with d=2:
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| <math>{\displaystyle}x^2 + y^2 = 1 + 2 x^2 y^2</math>
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| and the point P<sub>1</sub>=(1,0). The coordinates of the point 2P<sub>1</sub> are: | |
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| <math>x_3 = \frac{2x_1y_1}{1+dx_1^2y_1^2} = 0</math>
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| <math>y_3 = \frac{y_1^2-x_1^2}{1-dx_1^2y_1^2} = -1</math>
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| The point obtained from doubling P<sub>1</sub> is thus P<sub>3</sub>=(0,-1).
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| ==Mixed Addition==
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| Mixed addition is the case when Z<sub>2</sub> is known to be 1. In such a case A=Z<sub>1</sub>.Z<sub>2</sub> can be eliminated and the total cost reduces to 9'''M'''+1'''S'''+1'''C'''+1'''D'''+7'''a'''
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| ===Algorithm===
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| A= Z<sub>1</sub>.Z<sub>2</sub>
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| B= Z<sub>I</sub><sup>2</sup>
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| C=X<sub>1</sub>.X<sub>2</sub>
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| E=d.C.D
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| F=B-E
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| G=B+E
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| X<sub>3</sub>= Z<sub>1</sub>.F((X<sub>I</sub>+Y<sub>1</sub>).(X<sub>2</sub>+Y<sub>2</sub>)-C-D)
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| Y<sub>3</sub>= Z<sub>1</sub>.G.(D-C)
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| Z<sub>3</sub>=C.F.G
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| ==Tripling==
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| ''Tripling'' can be done by first doubling the point and then adding the result to itself. By applying the curve equation as in doubling, we obtain
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| : <math> 3(x_1,y_1) = \left( \frac{(x_1^2+ y_1^2) - (2 y_1)^2}{4(x_1^2-1)x_1^2 - (x_1^2-y_1^2)^2}x_1, \frac{(x_1^2+ y_1^2) - 2(x_1 )^2}{-4 (y_1^2-1)y_1^2+(x_1^2-y_1^2)^2}y_1 \right). \, </math>
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| There are two sets of formulas for this operation in standard Edwards coordinates. The first one costs 9'''M''' + 4'''S''' while the second needs 7'''M''' + 7'''S'''. If the '''S/M''' ratio is very small, specifically below 2/3, then the second set is better while for larger ratios the first one is to be preferred.<ref>Bernstein et al., Optimizing Double-Base Elliptic curve single-scalar Multiplication</ref>
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| Using the addition and doubling formulas (as mentioned above) the point (''X''<sub>1</sub> : ''Y''<sub>1</sub> : ''Z''<sub>1</sub>) is symbolically computed as 3(''X''<sub>1</sub> : ''Y''<sub>1</sub> : ''Z''<sub>1</sub>) and compared with (''X''<sub>3</sub> : ''Y''<sub>3</sub> : ''Z''<sub>3</sub>)
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| ;Example of tripling
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| Giving the Edwards curve with d=2, and the point P<sub>1</sub>=(1,0), the point 3P<sub>1</sub> has coordinates:
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| <math>x_3 = \frac{(x_1^2+ y_1^2) - (2 y_1)^2}{4(x_1^2-1)x_1^2 - (x_1^2-y_1^2)^2}x_1 = -1</math>
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| <math>y_3 = \frac{(x_1^2+ y_1^2 - 2(x_1 )^2}{-4 (y_1^2-1)y_1^2+(x_1^2-y_1^2)^2}y_1 = 0</math>
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| So, 3P<sub>1</sub>=(-1,0)=P-<sub>1</sub>. This result can also be found considering the doubling example: 2P<sub>1</sub>=(0,1), so 3P<sub>1</sub> = 2P<sub>1</sub> + P<sub>1</sub> = (0,-1) + P<sub>1</sub> = -P<sub>1</sub>.
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| ;Algorithm
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| A=X<sub>1</sub><sup>2</sup>
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| B=Y<sub>1</sub><sup>2</sup>
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| C=(2Z<sub>1</sub>)<sup>2</sup>
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| D=A+B
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| E=D<sup>2</sup>
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| F=2D.(A-B)
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| G=E-B.C
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| H=E-A.C
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| I=F+H
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| J=F-G
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| X<sub>3</sub>=G.J.X1
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| Y<sub>3</sub>=H.I.Y1
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| Z<sub>3</sub>=I.J.Z1
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| This formula costs 9'''M''' + 4'''S'''
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| ==Inverted Edwards coordinates==
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| Bernstein and Lange introduced an even faster coordinate system for elliptic curves called the ''Inverted Edward coordinates''<ref>Daniel J.Bernstein . Tanja Lange , pag.2,'' Inverted Edward coordinates''</ref> in which the coordinates (''X'' : ''Y'' : ''Z'') satisfy the curve (''X''<sup>2</sup> + ''Y''<sup>2</sup>)''Z''<sup>2</sup> = (''dZ''<sup>4</sup> + ''X''<sup>2</sup>''Y''<sup>2</sup>) and corresponds
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| to the affine point (''Z''/''X'', ''Z''/''Y'') on the Edwards curve ''x''<sup>2</sup> + ''y''<sup>2</sup> = 1 + ''dx''<sup>2</sup>''y''<sup>2</sup> with XYZ ≠ 0.
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| '''Inverted Edwards coordinates''', unlike standard Edwards coordinates, do not have complete addition formulas: some points, such as the neutral element, must be handled separately. But the addition formulas still have the advantage of strong unification: they can be used without change to double a point.
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| For more information about operations with these coordinates see http://hyperelliptic.org/EFD/g1p/auto-edwards-inverted.html
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| ==Extended Coordinates for Edward Curves==
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| There is another coordinates system with which an Edwards curve can be represented; these new coordinates are called '''extended coordinates'''<ref>H. Hisil, K. K. Wong, G. Carter, E. Dawson ''Faster group operations on elliptic curves''</ref> and are even faster than inverted coordinates. For more information about the time-cost required in the operations with these coordinates see:
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| http://hyperelliptic.org/EFD/g1p/auto-edwards.html
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| ==See also==
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| *[[Twisted Edwards curve]]
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| For more information about the running-time required in a specific case, see [[Table of costs of operations in elliptic curves]].
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| * {{Citation
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| |author=Daniel Bernstein, T.Lange
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| |year=2007
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| |title=''Faster Addition and Doubling on Elliptic curves''
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| |url=http://cr.yp.to/newelliptic/newelliptic-20070906.pdf
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| }}
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| * {{Citation | last1=Edwards | first1=Harold M. | title=A normal form for elliptic curves | url=http://www.ams.org/bull/2007-44-03/S0273-0979-07-01153-6/home.html | accessdate=2010-01-29 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | date=4/9/2007 | journal=[[Bulletin of the American Mathematical Society]] | issn=0002-9904 | volume=44 | pages=393–422}}
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| * ''Faster Group Operations on Elliptic curves'', H. Hisil, K. K. Wong, G. Carter, E. Dawson
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| * {{Citation
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| |author=D.J.Bernstein, P.Birkner. T. Lange, C.Peters
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| |title=''Optimizing Double-Base Elliptic-Curve Single-Scalar Multiplication''
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| |url=http://cr.yp.to/antiforgery/doublebase-20071028.pdf
| |
| }}
| |
| * {{Citation | last1=Washington | first1=Lawrence C. | title=Elliptic Curves: Number Theory and Cryptography | publisher=Chapman & Hall/CRC | edition=2nd | series=Discrete Mathematics and its Applications | isbn=978-1-4200-7146-7 | year=2008}}
| |
| *{{Citation
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| |author= Daniel J. Bernstein, T. Lange
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| |title= Inverted Edwards coordinates
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| |url=http://cr.yp.to/newelliptic/inverted-20071009.pdf
| |
| }}
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| ==External links==
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| * http://hyperelliptic.org/EFD/g1p/index.html
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| * http://hyperelliptic.org/EFD/g1p/auto-edwards.html
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| {{DEFAULTSORT:Edwards Curve}}
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| [[Category:Elliptic curves]]
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| [[Category:Elliptic curve cryptography]]
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