Addition of Elliptic Curves in Python

elliptic_curve.py

from fractions import Fraction
"""
    General form of the Curves:
    y**2 = x**3 + a*x**2 + b*x + c
    So the curve is defined by the tuple (a,b,c).
"""


def on_curve(x, y):
    """Check if (x,y) is on the curve y**2 == x**3+3*x"""
    return y ** 2 == x ** 3 + 3 * x


def sum1(cur, p1, p2):
   """compute p1+p2 on the elliptic curve cur
    :type p1: tuple
    """
    if p1 == p2:
        lam = (3*p1[0]**2 + 2 * cur[0] * p1[0] + cur[1])/Fraction(2*p1[1])
    elif p1[0] == p2[0]:
        return "null_element"
    else:
        lam = (p2[1] - p1[1]) / Fraction((p2[0] - p1[0]))
    
    nu = p1[1] - lam * p1[0]
    x3 = lam ** 2 - cur[0] - p1[0] - p2[0]
    y3 = -(lam * x3 + nu)
    
    """
        Because we have used Fraction on the denominator in order to turn it into a rational number,
        python is forced to perform a rational quotient so that __lam__ gets the proper value;
       
        Hence, getting this outcome __Fraction(-8, 9)__ merely means -8/9
    """
    return x3, y3


if __name__ == '__main__':
    
    curve1 = (0,0,17)
    p1_1=(-1,4)
    p2_1=(2,5)
    print(sum1(curve1, p1_1, p2_1))