Ćwiczenia 2

2/1.py

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+import gmpy2
+from gmpy2 import mpz, is_prime, powmod, invert, random_state, mpz_random
+import random
+
+# Task 1: Generate a random elliptic curve over F_p
+def generate_random_curve(p):
+    p = mpz(p)
+    rs = random_state(random.randrange(2**32))
+    while True:
+        A = mpz_random(rs, p)
+        B = mpz_random(rs, p)
+        # Check that the discriminant is non-zero
+        discriminant = (4 * powmod(A, 3, p) + 27 * powmod(B, 2, p)) % p
+        if discriminant != 0:
+            return A, B
+
+# Task 2: Find a random point on the elliptic curve over F_p
+def find_random_point(A, B, p):
+    p = mpz(p)
+    rs = random_state(random.randrange(2**32))
+    while True:
+        x = mpz_random(rs, p)
+        rhs = (powmod(x, 3, p) + A * x + B) % p
+        legendre_symbol = gmpy2.legendre(rhs, p)
+        if legendre_symbol == 1:
+            # Compute y using the fact that p ≡ 3 mod 4
+            y = powmod(rhs, (p + 1) // 4, p)
+            return x, y
+        elif legendre_symbol == 0:
+            # y = 0 is a valid solution
+            y = mpz(0)
+            return x, y
+        # If rhs is not a quadratic residue, try another x
+
+# Task 3: Compute the inverse (negative) of a given point
+def point_negation(P, p):
+    x, y = P
+    return x, (-y) % p
+
+# Task 4: Compute P ⊕ Q, the sum of two points on the elliptic curve
+def point_addition(P, Q, A, p):
+    if P == 'O':
+        return Q
+    if Q == 'O':
+        return P
+    x1, y1 = P
+    x2, y2 = Q
+    if x1 == x2 and (y1 + y2) % p == 0:
+        # P + (-P) = O
+        return 'O'
+    if x1 == x2 and y1 == y2:
+        # Point doubling
+        if y1 == 0:
+            return 'O'
+        numerator = (3 * powmod(x1, 2, p) + A) % p
+        denominator = (2 * y1) % p
+    else:
+        # P ≠ Q
+        numerator = (y2 - y1) % p
+        denominator = (x2 - x1) % p
+    inv_denominator = invert(denominator, p)
+    if inv_denominator is None:
+        return 'O'  # Cannot invert denominator; result is point at infinity
+    lam = (numerator * inv_denominator) % p
+    x3 = (powmod(lam, 2, p) - x1 - x2) % p
+    y3 = (lam * (x1 - x3) - y1) % p
+    return x3, y3
+
+# Task 5: Compute nP, the scalar multiplication of a point
+def scalar_multiplication(P, n, A, p):
+    n = mpz(n)
+    result = 'O'  # Start with the point at infinity
+    addend = P
+    while n > 0:
+        if n % 2 == 1:
+            result = point_addition(result, addend, A, p)
+        addend = point_addition(addend, addend, A, p)
+        n = n // 2
+    return result
+
+# Helper function to generate a large prime p ≡ 3 mod 4
+def generate_large_prime(bits):
+    rs = random_state(random.randrange(2**32))
+    while True:
+        candidate = mpz_random(rs, 2 ** bits)
+        candidate |= 1  # Ensure it's odd
+        candidate |= 3  # Ensure p ≡ 3 mod 4
+        candidate = gmpy2.next_prime(candidate)
+        if candidate % 4 == 3 and candidate.bit_length() == bits:
+            return candidate
+
+# Example usage of the functions
+if __name__ == "__main__":
+    # Generate a large prime p ≡ 3 mod 4 (approximately 300 bits)
+    bits = 300
+    p = generate_large_prime(bits)
+    print(f"Generated prime p = {p}")
+
+    # Task 1: Generate random curve coefficients A and B
+    A, B = generate_random_curve(p)
+    print(f"Random curve coefficients: A = {A}, B = {B}")
+
+    # Task 2: Find a random point on the curve
+    P = find_random_point(A, B, p)
+    print(f"Random point P = {P}")
+
+    # Task 3: Compute the inverse of point P
+    negative_P = point_negation(P, p)
+    print(f"Inverse of P: -P = {negative_P}")
+
+    # Task 4: Add two points P and Q on the curve
+    Q = find_random_point(A, B, p)
+    print(f"Another random point Q = {Q}")
+    R = point_addition(P, Q, A, p)
+    print(f"Sum of P and Q: R = P ⊕ Q = {R}")
+
+    # Task 5: Compute nP for a random scalar n
+    n = mpz_random(random_state(random.randrange(2**32)), p)
+    nP = scalar_multiplication(P, n, A, p)
+    print(f"Scalar multiplication: {n} * P = {nP}")