import math
import secrets
import numpy as np

# Constants
P = int("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F", 16)
Gx = int("79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798", 16)
Gy = int("483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8", 16)

# Utility Functions
def napier_log(n):
    """Custom logarithmic function based on Napier's concept."""
    return math.log(n, 2)  # Base-2 logarithm

def custom_summation(P, n=100):
    """Perform custom summation as per the NQSF formula."""
    # Use NumPy for vectorized summation
    j = np.arange(1, n + 1)
    terms = P ** j / (j ** 2 + 1)
    return np.sum(terms)

# Generate Keys
def generate_keys():
    """Generate a random private-public key pair."""
    private_key = secrets.randbelow(P - 1) + 1
    public_key_x = (Gx * private_key) % P
    public_key_y = (Gy * private_key) % P
    return private_key, (public_key_x, public_key_y)

# Compute Intermediates
def compute_intermediates(P, Q_G, G_Q):
    """Precompute reusable components of the formula."""
    a = Q_G ** P + G_Q ** P
    b = G_Q + Q_G
    c = napier_log(P)
    d = math.sin(P) * math.cos(P)
    e = P ** 4 + 2 * P ** 2 + 1
    f = P ** 2 + P + 1
    g = math.exp(G_Q)
    h = math.log(Q_G) / math.log(G_Q)
    i = custom_summation(P, 100)
    return a, b, c, d, e, f, g, h, i

# NQSF Formula
def nqsf_formula(P, Q_G, G_Q, intermediates):
    """Compute Napier Quantum Secure Formula."""
    a, b, c, d, e, f, g, h, i = intermediates
    # Simplified computation with cached values
    x = (a / (b * c + d)) * (e + f * g)
    x += Q_G ** P * (G_Q ** 2 + Q_G ** 2) + G_Q ** P * (Q_G ** 2 + G_Q ** 2)
    x += h
    x += P ** 3 + 2 * P ** 2 + 1
    x += i
    return x % P
