RSA 加密方案

RSA 算法

RSA 加密和签名：因大整数因子分解难算，合数可成公钥。

d - 私钥，e - 公钥，n - 可公开的合数，(e,n) 作为公钥可以公开，(d,n) 作为私钥。

详细理论证明参考：RSA算法原理（二）

假设明文消息为 M，密文为 C
加密过程： $M^e\mod{n}=C$
解密过程： $C^d\mod{n}=M$

RSA 签名举例

#-*- coding:utf-8 -*-
import random# 求最大公约数
def gcd(a, b):if a < b:return gcd(b, a)elif a % b == 0:return belse:return gcd(b, a % b)# 快速幂+取模
def power(a, b, c):ans = 1while b != 0:if b & 1:ans = (ans * a) % cb >>= 1a = (a * a) % creturn ans# 快速幂
def quick_power(a: int, b: int) -> int:ans = 1while b != 0:if b & 1:ans = ans * ab >>= 1a = a * areturn ans# 大素数检测
def Miller_Rabin(n):a = random.randint(2, n - 2)  # 随机第选取一个a∈[2,n-2]# print("随机选取的a=%lld\n"%a)s = 0  # s为d中的因子2的幂次数。d = n - 1while (d & 1) == 0:  # 将d中因子2全部提取出来。s += 1d >>= 1x = power(a, d, n)for i in range(s):  # 进行s次二次探测newX = power(x, 2, n)if newX == 1 and x != 1 and x != n - 1:return False  # 用二次定理的逆否命题，此时n确定为合数。x = newXif x != 1:  # 用费马小定理的逆否命题判断，此时x=a^(n-1) (mod n)，那么n确定为合数。return Falsereturn True  # 用费马小定理的逆命题判断。能经受住考验至此的数，大概率为素数。# 卢卡斯-莱墨素性检验
def Lucas_Lehmer(num: int) -> bool:  # 快速检验pow(2,m)-1是不是素数if num == 2:return Trueif num % 2 == 0:return Falses = 4Mersenne = pow(2, num) - 1  # pow(2, num)-1是梅森数for x in range(1, (num - 2) + 1):  # num-2是循环次数，+1表示右区间开s = ((s * s) - 2) % Mersenneif s == 0:return Trueelse:return False# 扩展的欧几里得算法，ab=1 (mod m), 得到a在模m下的乘法逆元b
def Extended_Eulid(a: int, m: int) -> int:def extended_eulid(a: int, m: int):if a == 0:  # 边界条件return 1, 0, melse:x, y, gcd = extended_eulid(m % a, a)  # 递归x, y = y, (x - (m // a) * y)  # 递推关系，左端为上层return x, y, gcd  # 返回第一层的计算结果。# 最终返回的y值即为b在模a下的乘法逆元# 若y为复数，则y+a为相应的正数逆元n = extended_eulid(a, m)if n[1] < 0:return n[1] + melse:return n[1]# 按照需要的 bit 来生成大素数
def Generate_prime(key_size: int) -> int:while True:num = random.randrange(quick_power(2, key_size - 1), quick_power(2, key_size))if Miller_Rabin(num): # 大概率是素数return num# 生成公钥和私钥
def KeyGen(p: int, q: int):n = p * qe = random.randint(1, (p - 1) * (q - 1))while gcd(e, (p - 1) * (q - 1)) != 1:e = random.randint(1, (p - 1) * (q - 1))d = Extended_Eulid(e, (p - 1) * (q - 1))return n, e, ddef Sign(x: int, d: int, n: int) -> int:s = power(x, d, n)return sdef Verify(s: int, e: int, n: int) -> int:x_ = power(s, e, n)return x_if __name__ == '__main__':key_size = 512p = Generate_prime(key_size)q = Generate_prime(key_size)n, e, d = KeyGen(p, q)# 消息x = int(input("Message: "))if type(x) != int: raise ValueError("Must be an integer!")# 签名s = Sign(x, d, n)# 验证x_ = Verify(s, e, n)Valid = (x_ == x)# Attacks_ = random.randint(1, (p - 1) * (q - 1))m_ = random.randint(1, (p - 1) * (q - 1))# 记录print("p:\t", p)print("q:\t", q)print("Private Key↓")print("N:\t", n)print("d:\t", d)print("Public Key↓")print("N:\t", n)print("e:\t", e)print("Signature↓")print("s:\t", s)