A twisted Edwards curve over a field 
of field characteristic
other than 2 is the affine plane curve
 |
(1)
|
where 
and 
.
The special case 
is an Edwards curve, whose underlying elliptic
curve normal form was introduced by Edwards (2007). Bernstein et al. (2008)
generalized it by allowing the coefficient 
not to be fixed to 1, giving the twisted Edwards curves above.
For points 
and 
,
the elliptic curve group law is 
, where
 |  |  |
(2)
|
 |  |  |
(3)
|
The identity element is 
, and the inverse element
of 
is 
.
When 
is a square and 
is a nonsquare in 
,
the addition formulas are complete (Bernstein et al. 2008).
