Edwards curve

In mathematics, the Edwards curves are a family of elliptic curves studied by 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 and Tanja Lange: they pointed out several advantages of the Edwards form in comparison to the more well known Weierstrass form.^[1]

The equation of an Edwards curve over a field K which does not have characteristic 2 is:

${\displaystyle x^{2}+y^{2}=1+d\ x^{2}y^{2}\,}$

for some scalar ${\displaystyle d\in K\setminus \{0,1\}}$. Also the following form with parameters c and d is called an Edwards curve:

${\displaystyle x^{2}+y^{2}=c^{2}(1+d\ x^{2}y^{2})\,}$

where c, d ∈ K with cd(1 − c⁴·d) ≠ 0.

Every Edwards curve is birationally equivalent to an elliptic curve in Montgomery form, and thus admits an algebraic group law once one chooses a point to serve as a neutral element. If K is finite, then a sizeable fraction of all elliptic curves over K can be written as Edwards curves. 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.

(See also Weierstrass curve group law)

Every Edwards curve ${\displaystyle x^{2}+y^{2}=1+dx^{2}y^{2}}$ over field K with characteristic not equal to 2 with ${\displaystyle d\neq 1}$ is birationally equivalent to an elliptic curve over the same field: ${\displaystyle (1/e)v^{2}=u^{3}+(4/e-2)u^{2}+u}$, where ${\displaystyle e=1-d,u={\frac {1+y}{1-y}},v={\frac {2(1+y)}{x(1-y)}}}$ and the point ${\displaystyle P=(0,1)}$ is mapped to the infinity O. This birational mapping induces a group on any Edwards curve.

An analogue on the circle

To understand better the concept of "addition" of points on a curve, a nice example is given by the classical circle group:

take the circle of radius 1

${\displaystyle {\displaystyle }x^{2}+y^{2}=1}$

and consider two points P₁=(x₁,y₁), P₂=(x₂,y₂) on it. Let α₁ and α₂ be the angles such that:

${\displaystyle {\displaystyle }P_{1}=(x_{1},y_{1})=(\sin {\alpha _{1}},\cos {\alpha _{1}})}$

${\displaystyle {\displaystyle }P_{2}=(x_{2},y_{2})=(\sin {\alpha _{2}},\cos {\alpha _{2}})}$

The sum of P₁ and P₂ is, thus, given by the sum of "their angles". That is, the point P₃=P₁+P₂ is a point on the circle with coordinates (x₃,y₃), where:

${\displaystyle {\displaystyle }x_{3}=\sin({\alpha }_{1}+{\alpha }_{2})=\sin {\alpha }_{1}\cos {\alpha }_{2}+\sin {\alpha }_{2}\cos {\alpha }_{1}=x_{1}y_{2}+x_{2}y_{1}}$

${\displaystyle {\displaystyle }y_{3}=\cos({\alpha }_{1}+{\alpha }_{2})=\cos {\alpha }_{1}\cos {\alpha }_{2}-\sin {\alpha }_{1}\sin {\alpha }_{2}=y_{1}y_{2}-x_{1}x_{2}.}$

In this way, the addition formula for points on the circle of radius 1 is:

${\displaystyle {\displaystyle }(x_{1},y_{1})+(x_{2},y_{2})=(x_{1}y_{2}+x_{2}y_{1},y_{1}y_{2}-x_{1}x_{2})}$.

Addition on Edwards curves

The points on an elliptic curve form an abelian group: one can add points and take integer multiples of a single point. When an elliptic curve is described by a non-singular cubic equation, then the sum of two points P and Q, denoted P + Q, is directly related to third point of intersection between the curve and the line that passes through P and Q.

The birational mapping between an Edwards curve and the corresponding cubic elliptic curve maps the straight lines into conic sections^[3] ${\displaystyle Axy+Bx+Cy+D=0}$. In other words, for the Edwards curves the three points ${\displaystyle P}$, ${\displaystyle Q}$ and ${\displaystyle -(P+Q)}$ lie on a hyperbola.

Given two distinct non-identity points ${\displaystyle P_{1}=(x_{1},y_{1}),P_{2}=(x_{2},y_{2}),P_{1}\neq P_{2}}$, the coefficients of the quadratic form are (up to scalars):

${\displaystyle A=(x_{1}-x_{2})+(x_{1}y_{2}-x_{2}y_{1})}$,

${\displaystyle B=(x_{2}y_{2}-x_{1}y_{1})+y_{1}y_{2}(x_{2}-x_{1})}$,

${\displaystyle C=x_{1}x_{2}(y_{1}-y_{2}),D=C}$

In the case of doubling a point ${\displaystyle P=(x,y)}$ the inverse point ${\displaystyle -2P}$ lies on the conic that touches the curve at the point ${\displaystyle P}$. The coefficients of the quadratic form that defines the conic are (up to scalars^{[clarification needed]}):

${\displaystyle A=dx^{2}y-1}$,

${\displaystyle B=y-x^{2}}$,

${\displaystyle C=x(1-y),D=C}$

In the context of cryptography, homogeneous coordinates are used to prevent 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 coordinates as:

${\displaystyle (X^{2}+Y^{2})Z^{2}=Z^{4}+dX^{2}Y^{2}}$.

A projective point ${\displaystyle (X:Y:Z)}$ corresponds to the affine point ${\displaystyle (X/Z:Y/Z)}$ on the Edwards curve.

The identity element is represented by ${\displaystyle (0:1:1)}$. The inverse of ${\displaystyle (X:Y:Z)}$ is ${\displaystyle (-X:Y:Z)}$.

The addition formula in homogeneous coordinates is given by: ${\displaystyle (X_{1}:Y_{1}:Z_{1})+(X_{2}:Y_{2}:Z_{2})=(X_{3}:Y_{3}:Z_{3})}$

where

${\displaystyle X_{3}=Z_{1}Z_{2}(X_{1}Y_{2}+X_{2}Y_{1})(Z_{1}^{2}Z_{2}^{2}-dX_{1}X_{2}Y_{1}Y_{2})}$

${\displaystyle Y_{3}=Z_{1}Z_{2}(Y_{1}Y_{2}-X_{1}X_{2})(Z_{1}^{2}Z_{2}^{2}+dX_{1}X_{2}Y_{1}Y_{2})}$

${\displaystyle Z_{3}=(Z_{1}^{2}Z_{2}^{2}-dX_{1}X_{2}Y_{1}Y_{2})(Z_{1}^{2}Z_{2}^{2}+dX_{1}X_{2}Y_{1}Y_{2})}$

Doubling can be performed with exactly the same formula as addition. Doubling refers to the case in which the inputs (x₁, y₁) and (x₂, y₂) are equal.

Doubling a point ${\displaystyle P=(x,y)}$:

${\displaystyle {\begin{aligned}(x,y)+(x,y)&=\left({\frac {2xy}{1+dx^{2}y^{2}}},{\frac {y^{2}-x^{2}}{1-dx^{2}y^{2}}}\right)\\[6pt]&=\left({\frac {2xy}{x^{2}+y^{2}}},{\frac {y^{2}-x^{2}}{2-x^{2}-y^{2}}}\right)\end{aligned}}}$

The denominators were simplified based on the curve equation ${\displaystyle x^{2}+y^{2}=1+dx^{2}y^{2}}$. Further speedup is achieved by computing ${\displaystyle 2xy}$ as ${\displaystyle (x+y)^{2}-x^{2}-y^{2}}$. This reduces the cost of doubling in homomorphic coordinates to 3M + 4S + 3C + 6a, while general addition costs 10M + 1S + 1C + 1D + 7a. Here M is field multiplications, S is field squarings, D is the cost of multiplying by the curve parameter d, and a is field addition.

Example of doubling

As in the previous example for the addition law, consider the Edwards curve with d=2:

${\displaystyle x^{2}+y^{2}=1+2x^{2}y^{2}}$

and the point ${\displaystyle P=(1,0)}$. The coordinates of the point ${\displaystyle P_{2}=2P_{1}}$ are:

${\displaystyle x_{2}={\frac {2xy}{x^{2}+y^{2}}}=0}$

${\displaystyle y_{2}={\frac {y^{2}-x^{2}}{2-(x^{2}+y^{2})}}=-1}$

The point obtained from doubling P is thus ${\displaystyle P_{2}=(0,-1)}$.

Mixed addition is the case when Z₂ is known to be 1. In such a case A=Z₁^.Z₂ can be eliminated and the total cost reduces to 9M+1S+1C+1D+7a

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

${\displaystyle 3(x_{1},y_{1})=\left({\frac {(x_{1}^{2}+y_{1}^{2})^{2}-(2y_{1})^{2}}{4(x_{1}^{2}-1)x_{1}^{2}-(x_{1}^{2}-y_{1}^{2})^{2}}}x_{1},{\frac {(x_{1}^{2}+y_{1}^{2})^{2}-(2x_{1})^{2}}{-4(y_{1}^{2}-1)y_{1}^{2}+(x_{1}^{2}-y_{1}^{2})^{2}}}y_{1}\right).\,}$

There are two sets of formulas for this operation in standard Edwards coordinates. The first one costs 9M + 4S while the second needs 7M + 7S. 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.^[5] Using the addition and doubling formulas (as mentioned above) the point (X₁ : Y₁ : Z₁) is symbolically computed as 3(X₁ : Y₁ : Z₁) and compared with (X₃ : Y₃ : Z₃)

Example of tripling

Giving the Edwards curve with d=2, and the point P₁=(1,0), the point 3P₁ has coordinates:

${\displaystyle x_{3}={\frac {(x_{1}^{2}+y_{1}^{2})^{2}-(2y_{1})^{2}}{4(x_{1}^{2}-1)x_{1}^{2}-(x_{1}^{2}-y_{1}^{2})^{2}}}x_{1}=-1}$

${\displaystyle y_{3}={\frac {(x_{1}^{2}+y_{1}^{2})^{2}-2(x_{1})^{2}}{-4(y_{1}^{2}-1)y_{1}^{2}+(x_{1}^{2}-y_{1}^{2})^{2}}}y_{1}=0}$

So, 3P₁=(-1,0)=P-₁. This result can also be found considering the doubling example: 2P₁=(0,1), so 3P₁ = 2P₁ + P₁ = (0,-1) + P₁ = -P₁.

Algorithm

A=X₁²

B=Y₁²

C=(2Z₁)²

D=A+B

E=D²

F=2D.(A-B)

G=E-B.C

H=E-A.C

I=F+H

J=F-G

X₃=G.J.X1

Y₃=H.I.Y1

Z₃=I.J.Z1

This formula costs 9M + 4S

Bernstein and Lange introduced an even faster coordinate system for elliptic curves called the Inverted Edward coordinates^[6] in which the coordinates (X : Y : Z) satisfy the curve (X² + Y²)Z² = (dZ⁴ + X²Y²) and corresponds to the affine point (Z/X, Z/Y) on the Edwards curve x² + y² = 1 + dx²y² with XYZ ≠ 0.

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.

For more information about operations with these coordinates see http://hyperelliptic.org/EFD/g1p/auto-edwards-inverted.html

There is another coordinates system with which an Edwards curve can be represented. These new coordinates are called extended coordinates^[7] and are even faster than inverted coordinates. For more information about the time-cost required in the operations with these coordinates see: http://hyperelliptic.org/EFD/g1p/auto-edwards.html

• Twisted Edwards curve

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