Edwards Curves

Synonyms

Edwards coordinates

Related Concepts

Elliptic Curve Cryptography; Elliptic Curve Method of Factorization; Elliptic Curves; Elliptic Curves for Primality Proving; Hyperelliptic Curves

Definition

An Edwards curve is an elliptic curve. Let k be a field in which $2\neq 0$ and let $d \in k\setminus \{0,1\}$; then ${x}^{2} + {y}^{2} = 1 + d{x}^{2}{y}^{2}$ defines an Edwards curve.

Background

Harold Edwards showed in [9] that any elliptic curve over a field in which $2\neq 0$ can be written in the form ${x}^{2} + {y}^{2} = {a}^{2}(1 + {x}^{2}{y}^{2})$, where ${a}^{4}\neq 1$ or 0, possibly after a finite field extension. Edwards also showed how to do arithmetic on this curve shape, i.e., how to add two points. In [6] Bernstein and Lange generalized the curve shape to ${x}^{2} + {y}^{2} = 1 + d{x}^{2}{y}^{2}$ with d ∈ k ∖ {0, 1}; these curves are now called Edwards curves. They also provided explicit formulas for adding and doubling points in projective coordinates....