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 Bernstein and Lange: they pointed out several advantages of the Edwards form in comparison to the more well known Weierstrass form.

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

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

where c, d ∈ K with cd(1 − c⁴·d) ≠ 0.

Every Edwards curve is birationally equivalent to an elliptic curve in Weierstrass 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.

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