We close this section with an explicit formula for
adding two points in .
If is an elliptic curve over a field ,
given by an equation
, then we
can compute the group addition using the following
algorithm.

Algorithm 10.1.2 (Elliptic Curve Group Law)
Given
,
this algorithm computes the sum
.

[One Point ] If set or if set
and terminate. Otherwise write
.

[Negatives] If and
, set and terminate.

[Compute ]Set
Note: If and , output and terminate.

[Compute Sum] Then
,
where
and is the coordinate of .