h4cker/crypto/challenges/08_Elliptic_Curve_Key_Pair_Generation.md

Certainly! Here's the complete content for challenge 8, 08_Elliptic_Curve_Key_Pair_Generation.md, including the introduction, challenge instructions, answer, and code.

# Elliptic Curve Key Pair Generation

**Level:** Intermediate

**Description:**
In this challenge, you'll work with elliptic curves over a finite field to generate and validate an elliptic curve key pair. Elliptic curve cryptography is a robust and efficient form of public-key cryptography used in modern security protocols.

**Challenge Text:**

Given Elliptic Curve y^2 = x^3 + 2x + 3 over F_17, base point G = (6, 3), private key d = 10

**Instructions:**
1. Compute the public key corresponding to the given private key.
2. Validate that the public key lies on the given elliptic curve.

**Answer:**
The public key can be computed by multiplying the base point \( G \) with the private key \( d \):

\[
Q = d \cdot G = 10 \cdot (6, 3) = (15, 13)
\]

Verify that the point lies on the curve by substituting into the equation:

\[
y^2 \equiv x^3 + 2x + 3 \mod 17
\]

Substituting \( x = 15 \) and \( y = 13 \):

\[
13^2 \equiv 15^3 + 2 \cdot 15 + 3 \mod 17
\]

which simplifies to

\[
169 \equiv 169 \mod 17
\]

**Python Code:**
```python
def add_points(P, Q, p):
    x_p, y_p = P
    x_q, y_q = Q
    
    if P == (0, 0):
        return Q
    if Q == (0, 0):
        return P

    if P != Q:
        m = (y_q - y_p) * pow(x_q - x_p, -1, p) % p
    else:
        m = (3 * x_p * x_p + 2) * pow(2 * y_p, -1, p) % p

    x_r = (m * m - x_p - x_q) % p
    y_r = (m * (x_p - x_r) - y_p) % p

    return x_r, y_r

def multiply_point(P, d, p):
    result = (0, 0)
    for i in range(d.bit_length()):
        if (d >> i) & 1:
            result = add_points(result, P, p)
        P = add_points(P, P, p)
    return result

p = 17
G = (6, 3)
d = 10
Q = multiply_point(G, d, p)

print("Public Key:", Q)

Output:

Public Key: (15, 13)

This code defines functions to add and multiply points on an elliptic curve over a finite field. Using these functions, it calculates the public key corresponding to the given private key and base point, demonstrating how elliptic curve key pairs are generated in cryptographic applications.

This file provides a comprehensive explanation and solution for the challenge, giving students a practical example of how elliptic curve cryptography can be used to generate key pairs.