Certainly, I can help you with that. First, let's define what we need:

Elliptic Curve Definition: An elliptic curve over a field (typically the real numbers or a finite field for cryptography) is given by the equation:
[
y^2 = x^3 + ax + b
]
where ( a ) and ( b ) are constants defining the curve, and we need to ensure ( 4a^3 + 27b^2 \neq 0 ) to avoid singular curves.

Point Addition: Given two points ( P = (x1, y1) ) and ( Q = (x2, y2) ) on the curve, the sum ( R = P + Q ) is defined as follows:
If ( P \neq Q ):
Calculate slope ( \lambda = \frac{y2 - y1}{x2 - x1} )
( x3 = \lambda^2 - x1 - x2 )
( y3 = \lambda(x1 - x3) - y1 )
If ( P = Q ) (point doubling):
( \lambda = \frac{3x1^2 + a}{2y1} )
( x3 = \lambda^2 - 2x1 )
( y3 = \lambda(x1 - x3) - y1 )
If one point is the point at infinity, the result is the other point.

Here's a Python implementation for this:

def add_elliptic_curve_points(a, b, x1, y1, x2, y2):
# Check if we're dealing with the point at infinity
if x1 is None and y1 is None:
return x2, y2
if x2 is None and y2 is None:
return x1, y1

# Point addition or doubling
if x1 == x2 and y1 == y2: # Doubling the point
if y1 == 0: # If y1 is 0, then P + P = O (point at infinity)
return None, None
# Calculate lambda for point doubling
lambda_val = (3 * x1 * x1 + a) / (2 * y1)
else: # Point addition
if x1 == x2: # Points are additive inverses, result is point at infinity
return None, None
# Calculate lambda for different points
lambda_val = (y2 - y1) / (x2 - x1)

# Common calculations for both cases
x3 = lambda_val**2 - x1 - x2
y3 = lambda_val * (x1 - x3) - y1

return (x3, y3)

1. Example usage

def example_usage():
# Define the curve y^2 = x^3 + ax + b
a, b = 1, 1 # Example coefficients for the curve
x1, y1 = 0, 1 # Point P
x2, y2 = 2, 3 # Point Q

# Adding different points
x3, y3 = add_elliptic_curve_points(a, b, x1, y1, x2, y2)
print(f"P + Q = ({x3}, {y3})")

# Doubling a point
x3, y3 = add_elliptic_curve_points(a, b, x1, y1, x1, y1)
print(f"P + P = ({x3}, {y3})")

# Adding point with its inverse should return None, None (point at infinity)
x3, y3 = add_elliptic_curve_points(a, b, x1, y1, x1, -y1 % (10**9 + 7)) # Assuming mod for finite field
print(f"P + (-P) = ({x3}, {y3})")

1. Run the example

example_usage()

Notes:

This function assumes we're working over the real numbers for simplicity. For cryptographic applications, you'd work over a finite field, where addition, subtraction, multiplication, and division (inverses) are done modulo some prime ( p ).
The modulus operation in the example usage when adding a point with its inverse suggests how you might handle this in a finite field context, although for real-world crypto, you'd use a proper finite field library.
The example does not check for the condition ( 4a^3 + 27b^2 \neq 0 ), which should be done when defining the curve in practice.