Create 08_Elliptic_Curve_Key_Pair_Generation.md

crypto/challenges/08_Elliptic_Curve_Key_Pair_Generation.md

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