In algebraic geometry, the twisted Edwards curves are plane models of elliptic curves, a generalisation of Edwards curves introduced by Bernstein, Birkner, Joye, Lange and Peters in 2008. 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 a digital signature scheme called EdDSA that offers high performance while avoiding security problems that have surfaced in other digital signature schemes.
Definition A twisted Edwards curve E E , a , d {\displaystyle E_{E,a,d}} over a field K {\displaystyle \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 + d x 2 y 2 {\displaystyle E_{E,a,d}:ax^{2}+y^{2}=1+dx^{2}y^{2}}
where a , d {\displaystyle a,d} are distinct non-zero elements of K {\displaystyle \mathbb {K} } . Each twisted Edwards curve is a twist of an Edwards curve. The special case a = 1 {\displaystyle 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 K {\displaystyle \mathbb {K} } be a field with characteristic different from 2. Let ( x 1 , y 1 ) {\displaystyle (x_{1},y_{1})} and ( x 2 , y 2 ) {\displaystyle (x_{2},y_{2})} be points on the twisted Edwards curve. The equation of twisted Edwards curve is written as;
E E , a , d {\displaystyle E_{E,a,d}} : a x 2 + y 2 = 1 + d x 2 y 2 {\displaystyle ax^{2}+y^{2}=1+dx^{2}y^{2}} . The sum of these points ( x 1 , y 1 ) , ( x 2 , y 2 ) {\displaystyle (x_{1},y_{1}),(x_{2},y_{2})} on E E , a , d {\displaystyle E_{E,a,d}} is:
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