A Python script generate the Elliptic-curve chart.

ecc.py

import numpy as np
import matplotlib.pyplot as plt

def modular_sqrt(a, p):
    """
    Compute the modular square root of a modulo p using the Tonelli-Shanks algorithm.
    Returns the two square roots if they exist, or None if no square root exists.
    """
    a = int(a)  # Convert a to a regular integer
    if pow(a, (p-1)//2, p) != 1:
        return None
    q = p - 1
    s = 0
    while q % 2 == 0:
        q //= 2
        s += 1
    if s == 1:
        return pow(a, (p+1)//4, p)
    for z in range(2, p):
        if p - 1 == pow(z, (p-1)//2, p):
            break
    c = pow(z, q, p)
    r = pow(a, (q+1)//2, p)
    t = pow(a, q, p)
    m = s
    while (t - 1) % p != 0:
        t2 = (t * t) % p
        for i in range(1, m):
            if (t2 - 1) % p == 0:
                break
            t2 = (t2 * t2) % p
        b = pow(c, 1 << (m - i - 1), p)
        r = (r * b) % p
        c = (b * b) % p
        t = (t * c) % p
        m = i
    return r, p-r

# Define the elliptic curve parameters
p = 13  # Modulus
a = 5   # Coefficient of x
b = 9   # Constant term

# Generate x values
x = np.arange(0, p)

# Calculate y values
y = []
x_coords = []
for xi in x:
    rhs = (int(xi)**3 + a*int(xi) + b) % p
    sqrt_rhs = modular_sqrt(rhs, p)
    if sqrt_rhs:
        y.append(sqrt_rhs[0])
        y.append(sqrt_rhs[1])
        x_coords.append(xi)
        x_coords.append(xi)

sorted_points = sorted(zip(x_coords, y), key=lambda x: x[0])

print("The points on the curve are:")
for i, point in enumerate(sorted_points):
    if i % 2 == 0:
        print("")
    print(point, end=", ")

# Define the points G and H
G = (7, 7)
H = (12, 9)

# Plot the elliptic curve
plt.figure(figsize=(6, 6))
plt.scatter(x_coords, y, label='Elliptic Curve', color='blue', marker='.')

# Plot the points G and H
plt.scatter(G[0], G[1], label='G (7, 7)', color='red', marker='o', s=100)
plt.scatter(H[0], H[1], label='H (12, 9)', color='green', marker='o', s=100)

plt.xlabel('X')
plt.ylabel('Y')
plt.title(f'Elliptic Curve over F_{p}: y^2 = x^3 + {a}x + {b} (mod {p})')
plt.legend()
plt.grid(True)
plt.xticks(range(0, p))
plt.yticks(range(0, p))
plt.gca().set_aspect('equal', adjustable='box')
plt.show()