Use Python to implement RSA encryption and decryption

I saw an English article [1] yesterday, showing how to use Python to implement the RSA algorithm. The logic of the code is the same as the previous article Understanding the RSA Algorithm. Friends who are not familiar with RSA can read the article Understanding the RSA Algorithm. , which explains what RSA is, the mathematical principles of RSA, and gives a simple example. It can be said to be the easiest article to understand RSA on Quanzhihu (this comes from a reader's comment).

I ran the code provided in English and found that it could not encrypt Chinese, so I modified the encryption and decryption functions to support Chinese encryption and decryption. Today’s article will share how to use Python to implement the RSA encryption and decryption process to help you establish an intuitive understanding of RSA. The random prime number generation algorithm in the code is also worth learning.

0. Effect demonstration

Let’s take a look at the effect first.

Original text: "There is a mole, terminate the transaction"

The cipher text cannot be cracked at all:

After decryption:

The complete code public account "Python No. 7" can be obtained by replying "rsa".

1. Key pair generation

Ideas:

1) Randomly find two prime numbers (prime numbers) p and q. The larger p and q, the safer they are. Here we choose a 1024-bit prime number:

p = genprime(1024)
q = genprime(1024)

genprime() Let’s not talk about the implementation process of the function.

2) Calculate their product n = p * q and Euler function lambda_n.

n = p * q
lambda_n = (p - 1) * (q - 1)

3) Randomly select an integer e, the condition is 1 < e < lambda_n, and e and lambda_n are relatively prime. For example, if you select 35537, 35537 has only 16 bits and must be smaller than lambda_n. < e < lambda_n，且 e 与 lambda_n 互质。比如选择 35537，35537 只有 16 位，必然小于 lambda_n。

e = 35537

4) Find an integer d such that the remainder of e * d divided by lambda_n is 1, and return the key pair.

d = eucalg(e, lambda_n)[0]
if d < 0: d += lambda_n
return (d, n), (e, n)

The implementation of the eucalg function will be discussed later.

At this point, the key pair generation function is as follows:

def create_keys():
 p = genprime(1024)
 q = genprime(1024)
 n = p * q
 lambda_n = (p - 1) * (q - 1)
 e = 35537
 d = eucalg(e, lambda_n)[0]
 if d < 0: d += lambda_n
 return (d, n), (e, n)

2. Implementation of encryption and decryption

The encryption and decryption processes are the same, public key encryption, private key encryption Key decryption, and vice versa, private key encryption and public key decryption, but the former is called encryption and the latter is called signature.

The specific function implementation is as follows:

def encrypt_data(data,key):
e_data = []
for d in data:
 e = modpow(d, key[0], key[1]) 
 e_data.append(e)
return e_data

## 加密和解密的逻辑完全一样
decrypt_data = encrypt_data

The modpow function is used here, which is used to calculate the formula b^e % n = r.

• If it is an encryption process, then b is the plaintext, (n,e) is the public key, and r is the ciphertext.
• If it is a decryption process, then b is the ciphertext, (n, d) is the private key, and r is the famous text.

modpow is defined as follows:

def modpow(b, e, n):
 # find length of e in bits
 tst = 1
 siz = 0
 while e >= tst:
tst <<= 1
siz += 1
 siz -= 1
 # calculate the result
 r = 1
 for i in range(siz, -1, -1):
r = (r * r) % n
if (e >> i) & 1: r = (r * b) % n
 return r

3. Generating function of random prime numbers

Generating function of random prime numbers, which uses matrix multiplication and Fibonacci Sequence shows the importance of mathematics to algorithms.

# matrix multiplication
def sqmatrixmul(m1, m2, w, mod):
 mr = [[0 for j in range(w)] for i in range(w)]
 for i in range(w):
for j in range(w):
 for k in range(w):
mr[i][j] = (mr[i][j] + m1[i][k] * m2[k][j]) % mod
 return mr

# fibonacci calculator
def fib(x, mod):
 if x < 3: return 1
 x -= 2
 # find length of e in bits
 tst = 1
 siz = 0
 while x >= tst:
tst <<= 1
siz += 1
 siz -= 1
 # calculate the matrix
 fm = [
# function matrix
[0, 1],
[1, 1]
 ]
 rm = [
# result matrix
# (identity)
[1, 0],
[0, 1]
 ]
 for i in range(siz, -1, -1):
rm = sqmatrixmul(rm, rm, 2, mod)
if (x >> i) & 1:
 rm = sqmatrixmul(rm, fm, 2, mod)

 # second row of resulting vector is result
 return (rm[1][0] + rm[1][1]) % mod

def genprime(siz):
 while True:
num = (1 << (siz - 1)) + secrets.randbits(siz - 1) - 10;
# num must be 3 or 7 (mod 10)
num -= num % 10
num += 3 # 3 (mod 10)
# heuristic test
if modpow(2, num - 1, num) == 1 and fib(num + 1, num) == 0:
 return num
num += 5 # 7 (mod 10)
# heuristic test
if modpow(2, num - 1, num) == 1 and fib(num + 1, num) == 0:
 return num

4. Implementation of eucalg function

The essence of the function is to find the solution to the following linear equation of two variables:

e * x - lambda_n * y =1

Specific code:

def eucalg(a, b):
 # make a the bigger one and b the lesser one
 swapped = False
 if a < b:
a, b = b, a
swapped = True
 # ca and cb store current a and b in form of
 # coefficients with initial a and b
 # a' = ca[0] * a + ca[1] * b
 # b' = cb[0] * a + cb[1] * b
 ca = (1, 0)
 cb = (0, 1)
 while b != 0:
# k denotes how many times number b
# can be substracted from a
k = a // b
# swap a and b so b is always the lesser one
a, b, ca, cb = b, a-b*k, cb, (ca[0]-k*cb[0], ca[1]-k*cb[1])
 if swapped:
return (ca[1], ca[0])
 else:
return ca

5 , test

test.py script usage method:

1), generate key

python test.py make-keys rsakey

The public key is saved in rsakey.pub, and the private key is saved in rsakey.priv中

2), Encrypt the file content

If there is a plain text .txt file:

python test.py encrypt 明文.txt from rsakey to 密文.txt

will generate a ciphertext .txt

3. Encrypt the file Content decryption

If there is a file ciphertext.txt:

python test.py decrypt 密文.txt as rsakey to 解密后.txt

will generate a decrypted .txt

Final words

This article shares the Python of the RSA algorithm A simple implementation can help understand the RSA algorithm.

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