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# 使用Python实现简单的RSA加密算法 RSA是一种非对称加密算法，下面我将实现一个简化版的RSA加密解密过程，并解释其原理。 ```python import random import math from typing import Tuple class SimpleRSA: def __init__(self, key_size=64): """ 初始化RSA，生成密钥对 key_size: 密钥大小（位），实际使用中至少需要1024位，这里为了演示使用小值 """ self.key_size = key_size # 生成两个大素数 p = self._generate_prime(key_size // 2) q = self._generate_prime(key_size // 2) # 计算n和φ(n) self.n = p * q phi = (p - 1) * (q - 1) # 选择公钥指数e self.e = self._choose_public_exponent(phi) # 计算私钥指数d self.d = self._mod_inverse(self.e, phi) print(f"生成的素数: p={p}, q={q}") print(f"n = {self.n}") print(f"φ(n) = {phi}") print(f"公钥: (e={self.e}, n={self.n})") print(f"私钥: (d={self.d}, n={self.n})") def _is_prime(self, n: int, k=10) -> bool: """Miller-Rabin素性测试""" if n < 2: return False if n in (2, 3): return True if n % 2 == 0: return False # 将n-1写成2^r * d的形式 r, d = 0, n - 1 while d % 2 == 0: r += 1 d //= 2 # 进行k次测试 for _ in range(k): a = random.randint(2, n - 2) x = pow(a, d, n) if x == 1 or x == n - 1: continue for _ in range(r - 1): x = pow(x, 2, n) if x == n - 1: break else: return False return True def _generate_prime(self, bits: int) -> int: """生成指定位数的大素数""" while True: # 生成一个奇数 num = random.getrandbits(bits) num |= (1 << bits - 1) | 1 # 确保最高位和最低位都是1 if self._is_prime(num): return num def _gcd(self, a: int, b: int) -> int: """欧几里得算法求最大公约数""" while b != 0: a, b = b, a % b return a def _choose_public_exponent(self, phi: int) -> int: """选择公钥指数e，通常为65537""" # 常用值65537 common_e = 65537 if common_e < phi and self._gcd(common_e, phi) == 1: return common_e # 否则选择一个较小的素数 e = 3 while self._gcd(e, phi) != 1: e += 2 return e def _extended_gcd(self, a: int, b: int) -> Tuple[int, int, int]: """扩展欧几里得算法""" if b == 0: return a, 1, 0 gcd, x1, y1 = self._extended_gcd(b, a % b) x = y1 y = x1 - (a // b) * y1 return gcd, x, y def _mod_inverse(self, a: int, m: int) -> int: """求模逆元""" gcd, x, _ = self._extended_gcd(a, m) if gcd != 1: raise ValueError(f"{a}在模{m}下没有逆元") return x % m def encrypt(self, message: int) -> int: """加密：c = m^e mod n""" if message >= self.n: raise ValueError("消息必须小于n") return pow(message, self.e, self.n) def decrypt(self, ciphertext: int) -> int: """解密：m = c^d mod n""" return pow(ciphertext, self.d, self.n) def encrypt_string(self, text: str) -> list: """加密字符串""" # 将字符串转换为数字 encoded = text.encode('utf-8') # 将字节转换为大整数 m = int.from_bytes(encoded, 'big') # 如果消息太长，需要分块（简化版，实际需要更复杂的分块机制） if m >= self.n: raise ValueError("消息太长，需要分块加密") ciphertext = self.encrypt(m) return ciphertext def decrypt_string(self, ciphertext: int) -> str: """解密为字符串""" m = self.decrypt(ciphertext) # 将整数转换回字节 byte_length = (m.bit_length() + 7) // 8 decoded = m.to_bytes(byte_length, 'big') return decoded.decode('utf-8') def rsa_example(): """RSA算法示例""" print("=" * 60) print("RSA非对称加密算法示例") print("=" * 60) # 创建RSA实例（使用小密钥以便演示） rsa = SimpleRSA(key_size=32) print("\n" + "-" * 60) # 示例1：加密数字 print("\n1. 加密数字:") original_number = 42 print(f"原始数字: {original_number}") encrypted = rsa.encrypt(original_number) print(f"加密后: {encrypted}") decrypted = rsa.decrypt(encrypted) print(f"解密后: {decrypted}") # 示例2：加密字符串 print("\n2. 加密字符串:") original_text = "Hello RSA!" print(f"原始文本: '{original_text}'") try: encrypted_text = rsa.encrypt_string(original_text) print(f"加密后: {encrypted_text}") decrypted_text = rsa.decrypt_string(encrypted_text) print(f"解密后: '{decrypted_text}'") except ValueError as e: print(f"错误: {e}") print("尝试更短的字符串...") original_text = "RSA" print(f"\n新文本: '{original_text}'") encrypted_text = rsa.encrypt_string(original_text) print(f"加密后: {encrypted_text}") decrypted_text = rsa.decrypt_string(encrypted_text) print(f"解密后: '{decrypted_text}'") # 示例3：验证加密解密过程 print("\n3. 验证RSA原理:") print(f"公钥(e, n): ({rsa.e}, {rsa.n})") print(f"私钥(d, n): ({rsa.d}, {rsa.n})") test_message = 123 print(f"\n测试消息: {test_message}") # 加密: c = m^e mod n cipher = pow(test_message, rsa.e, rsa.n) print(f"加密: {test_message}^{rsa.e} mod {rsa.n} = {cipher}") # 解密: m = c^d mod n decrypted = pow(cipher, rsa.d, rsa.n) print(f"解密: {cipher}^{rsa.d} mod {rsa.n} = {decrypted}") # 验证: m^(e*d) mod n = m verification = pow(test_message, rsa.e * rsa.d, rsa.n) print(f"验证: {test_message}^({rsa.e}*{rsa.d}) mod {rsa.n} = {verification}") print(f"验证结果: {'成功' if verification == test_message else '失败'}") def rsa_math_explanation(): """RSA数学原理解释""" print("\n" + "=" * 60) print("RSA算法数学原理") print("=" * 60) print("\n1. 密钥生成:") print(" a. 选择两个大素数 p 和 q") print(" b. 计算 n = p × q") print(" c. 计算 φ(n) = (p-1) × (q-1)") print(" d. 选择整数 e，使得

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