Recall from the discussion in Group Theory, we learnt how a generator point can be added to itself repeatedly to generate every element of the group. In this section, we'll understand how to perform this addition, and implement it in Python.

$P$
and
$Q$
on an elliptic curve, find the third point
$R$
where line joining
$P$
and
$Q$
intersects. This value of
$R$
is equal to
$-(P+Q)$
. Reflecting the point along the X-axis will give us
$P+Q$
.
Addition of two points on an elliptic curve over a field of real numbers.
To find the coordinates of the third point of intersection, simply calculate the slope between P and Q, and extrapolate it using the general equation of elliptic curve.
Addition of two points on an elliptic curve over a finite field.

Implementation in Python

inf = float("inf")
class Point:
... # add these methods to the previously defined Point class
#################################################################
# Point Addition for P₁ or P₂ = I (identity) #
# #
# Formula: #
# P + I = P #
# I + P = P #
#################################################################
if self == I:
return other
if other == I:
return self
#################################################################
# #
# Formula: #
# P + (-P) = I #
# (-P) + P = I #
#################################################################
if self.x == other.x and self.y == (-1 * other.y):
return I
#################################################################
# Point Addition for X₁ ≠ X₂ (line with slope) #
# #
# Formula: #
# S = (Y₂ - Y₁) / (X₂ - X₁) #
# X₃ = S² - X₁ - X₂ #
# Y₃ = S(X₁ - X₃) - Y₁ #
#################################################################
if self.x != other.x:
x1, x2 = self.x, other.x
y1, y2 = self.y, other.y
s = (y2 - y1) / (x2 - x1)
x3 = s ** 2 - x1 - x2
y3 = s * (x1 - x3) - y1
return self.__class__(
x=x3.value,
y=y3.value,
curve=secp256k1
)
#################################################################
# Point Addition for P₁ = P₂ (vertical tangent) #
# #
# Formula: #
# S = ∞ #
# (X₃, Y₃) = I #
#################################################################
if self == other and self.y == inf:
return I
#################################################################
# Point Addition for P₁ = P₂ (tangent with slope) #
# #
# Formula: #
# S = (3X₁² + a) / 2Y₁ .. ∂(Y²) = ∂(X² + aX + b) #
# X₃ = S² - 2X₁ #
# Y₃ = S(X₁ - X₃) - Y₁ #
#################################################################
if self == other:
x1, y1, a = self.x, self.y, self.curve.a
s = (3 * x1 ** 2 + a) / (2 * y1)
x3 = s ** 2 - 2 * x1
y3 = s * (x1 - x3) - y1
return self.__class__(
x=x3.value,
y=y3.value,
curve=secp256k1
)
Point at Infinity
Also known as the identity point, it is the third point where P and Q meet, in the figure below.
$P + (-P) = I$
Point at infinity is the third point where the line joining P and Q meets the curve.
We can initialise the point at infinity like this:
I = Point(x=None, y=None, curve=secp256k1)