一、RSA算法概述
rsa算法是一种非对称加密算法，其安全性是建立在大素数难以分解的基础上的，即将两个大素数相乘十分容易，但想对其乘积进行分解却很困难，所以可以将其乘积公开作为加密密钥
二、RSA算法设计理念
根据数论，寻求两个大素数比较简单，而将它们的乘积进行因式分解却极其困难，因此可以将乘积公开作为加密密钥
三、加解密过程及密钥生成
1、加解密过程
此处从明文和密文加密和解密开始，然后讲密钥的生成
(1). 对于明文M，则有密文C=M^e mod n (获得密文是明文的e次方再模n，即求余数)
(2). 对于密文C，则有明文M=C^d mod n (获得明文是密文的d次方再模n，即求余数)
明文和密文的产生是建立在一对密钥的基础上的，即(e,n)和(d,n) ,(e,n)称为公钥 , (d,n)称为私钥 (先记下公钥和私钥的概念，有个印象)
下面是一个形象的例子
假设A要与B通信：
A————————————————————————————B
(e,n) (d,n)
A握着（e,n）对想发送的明文M加密C=M^e mod n形成密文C，再将C发送给B
B拿到密文C，再用自己的私钥（d,n）对密文C解密还原明文M
现在我们只需要知道(e,n)和(d,n)即（e，d，n）三个密钥怎么来的就搞定了RSA算法
2、密钥生成过程 (e,d,n)
(1).求n
准备两个素数p，q（最好准备较大的素数） (注:素数 质数是同一个东东)
n=p*q
至此n得到了
(2).根据第一步准备的p和q计算 n的欧拉函数φ(n)
φ(n)=(p-1)*(q-1)
(3).选取公钥e
选取条件：质数，1<e<φ(n) , (e,φ(n))=1(e与φ(n)互质)
至此e得到了，在实际应用中，e一般为65537，(ctfer应该比较敏感吧hhh
(4).计算私钥d，计算e对于φ(n)的模反元素d。
d应满足：ed ≡ 1 (mod φ(n)) （即 （d*e）mod φ(n)=1）
至此(e,d,n)全部得出
四、python实现
明白了算法的原理，代码实现也就变的简单了
具体思路就是，按照p，q得到密钥e，d，n后，执行加密和解密的式子。
import random
'''
Euclid's algorithm for determining the greatest common divisor
Use iteration to make it faster for larger integers
'''
def gcd(a, b):
while b != 0:
a, b = b, a % b
return a
'''
Euclid's extended algorithm for finding the multiplicative inverse of two numbers
'''
def multiplicative_inverse(e, phi):
d = 0
x1 = 0
x2 = 1
y1 = 1
temp_phi = phi
while e > 0:
temp1 = temp_phi//e
temp2 = temp_phi - temp1 * e
temp_phi = e
e = temp2
x = x2 - temp1 * x1
y = d - temp1 * y1
x2 = x1
x1 = x
d = y1
y1 = y
if temp_phi == 1:
return d + phi
'''
Tests to see if a number is prime.
'''
def is_prime(num):
if num == 2:
return True
if num < 2 or num % 2 == 0:
return False
for n in range(3, int(num**0.5)+2, 2):
if num % n == 0:
return False
return True
def generate_key_pair(p, q):
if not (is_prime(p) and is_prime(q)):
raise ValueError('Both numbers must be prime.')
elif p == q:
raise ValueError('p and q cannot be equal')
# n = pq
n = p * q
# Phi is the totient of n
phi = (p-1) * (q-1)
# Choose an integer e such that e and phi(n) are coprime
e = random.randrange(1, phi)
# Use Euclid's Algorithm to verify that e and phi(n) are coprime
g = gcd(e, phi)
while g != 1:
e = random.randrange(1, phi)
g = gcd(e, phi)
# Use Extended Euclid's Algorithm to generate the private key
d = multiplicative_inverse(e, phi)
# Return public and private key_pair
# Public key is (e, n) and private key is (d, n)
return ((e, n), (d, n))
def encrypt(pk, plaintext):
# Unpack the key into it's components
key, n = pk
# Convert each letter in the plaintext to numbers based on the character using a^b mod m
cipher = [pow(ord(char), key, n) for char in plaintext]
# Return the array of bytes
return cipher
def decrypt(pk, ciphertext):
# Unpack the key into its components
key, n = pk
# Generate the plaintext based on the ciphertext and key using a^b mod m
aux = [str(pow(char, key, n)) for char in ciphertext]
# Return the array of bytes as a string
plain = [chr(int(char2)) for char2 in aux]
return ''.join(plain)
if __name__ == '__main__':
'''
Detect if the script is being run directly by the user
'''
print("===========================================================================================================")
print("================================== RSA Encryptor / Decrypter ==============================================")
print(" ")
p = int(input(" - Enter a prime number (17, 19, 23, etc): "))
q = int(input(" - Enter another prime number (Not one you entered above): "))
print(" - Generating your public / private key-pairs now . . .")
public, private = generate_key_pair(p, q)
print(" - Your public key is ", public, " and your private key is ", private)
message = input(" - Enter a message to encrypt with your public key: ")
encrypted_msg = encrypt(public, message)
print(" - Your encrypted message is: ", ''.join(map(lambda x: str(x), encrypted_msg)))
print(" - Decrypting message with private key ", private, " . . .")
print(" - Your message is: ", decrypt(private, encrypted_msg))
print(" ")
print("============================================ END ==========================================================")
print("===========================================================================================================")
