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August 3, 2024 04:45
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ElGamal Encryption and Digital Signatures
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| # ElGamal encryption | |
| import random | |
| from sympy import isprime, mod_inverse | |
| # Generate a large prime number for the modulus (p) | |
| def generate_large_prime(bits=256): | |
| while True: | |
| p = random.getrandbits(bits) | |
| if isprime(p): | |
| return p | |
| # Generate the keys for ElGamal encryption | |
| def generate_keys(bits=256): | |
| p = generate_large_prime(bits) | |
| g = random.randint(2, p - 1) | |
| x = random.randint(1, p - 2) # Private key | |
| h = pow(g, x, p) # Public key component | |
| return (p, g, h), x | |
| # Encrypt a message using the ElGamal encryption scheme | |
| def encrypt(public_key, plaintext): | |
| p, g, h = public_key | |
| y = random.randint(1, p - 2) | |
| c1 = pow(g, y, p) | |
| s = pow(h, y, p) | |
| c2 = (plaintext * s) % p | |
| return c1, c2 | |
| # Decrypt a ciphertext using the ElGamal encryption scheme | |
| def decrypt(private_key, public_key, ciphertext): | |
| p, g, h = public_key | |
| x = private_key | |
| c1, c2 = ciphertext | |
| s = pow(c1, x, p) | |
| s_inv = mod_inverse(s, p) | |
| plaintext = (c2 * s_inv) % p | |
| return plaintext | |
| # Example usage | |
| if __name__ == "__main__": | |
| public_key, private_key = generate_keys() | |
| print("Public Key:", public_key) | |
| print("Private Key:", private_key) | |
| plaintext = 0x12345678 | |
| print("Plaintext:", hex(plaintext)) | |
| ciphertext = encrypt(public_key, plaintext) | |
| print("Ciphertext:", ciphertext) | |
| decrypted_plaintext = decrypt(private_key, public_key, ciphertext) | |
| print("Decrypted Plaintext:", hex(decrypted_plaintext)) |
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| # ElGamal signatures | |
| from sympy import isprime, randprime, gcd, mod_inverse | |
| import random | |
| import hashlib | |
| def generate_prime(bits): | |
| return randprime(2**(bits-1), 2**bits) | |
| def find_generator(p): | |
| for g in range(2, p): | |
| if pow(g, (p-1)//2, p) != 1: | |
| return g | |
| return None | |
| def H(message, p): | |
| return int(hashlib.sha256(message).hexdigest(), 16) % (p-1) | |
| tau = 256 | |
| p = generate_prime(tau) | |
| g = find_generator(p) | |
| alpha = random.randint(1, p-2) | |
| y = pow(g, alpha, p) | |
| public_key = (p, g, y, lambda m: H(m, p)) | |
| private_key = (p, g, alpha, lambda m: H(m, p)) | |
| def pick_k(p): | |
| k = random.randint(1, p-2) | |
| while gcd(k, p-1) != 1: | |
| k = random.randint(1, p-2) | |
| return k | |
| def sign_message(m, private_key): | |
| p, g, alpha, H = private_key | |
| k = pick_k(p) | |
| r = pow(g, k, p) | |
| h_mr = H(m.encode() + str(r).encode()) | |
| k_inv = mod_inverse(k, p-1) | |
| s = (k_inv * (h_mr - alpha * r)) % (p-1) | |
| return (r, s) | |
| def verify_signature(m, signature, public_key): | |
| p, g, y, H = public_key | |
| r, s = signature | |
| if not (1 <= r < p): | |
| return False | |
| v = (pow(y, r, p) * pow(r, s, p)) % p | |
| h_mr = H(m.encode() + str(r).encode()) | |
| if v == pow(g, h_mr, p): | |
| return True | |
| else: | |
| return False | |
| def print_keys(public_key, private_key): | |
| p, g, y, _ = public_key | |
| _, _, alpha, _ = private_key | |
| print(f"Public Key:\np: {p}\ng: {g}\ny: {y}\n") | |
| print(f"Private Key:\np: {p}\ng: {g}\nalpha: {alpha}\n") | |
| # Generate keys | |
| public_key = (p, g, y, lambda m: H(m, p)) | |
| private_key = (p, g, alpha, lambda m: H(m, p)) | |
| # Print keys | |
| print_keys(public_key, private_key) | |
| # Sign and verify a message | |
| message = "Hello, World!" | |
| signature = sign_message(message, private_key) | |
| is_valid = verify_signature(message, signature, public_key) | |
| print(f"Signature valid: {is_valid}") | |
| print(f"Signature: {signature}") |
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