1. Generation of public and private keys:
- (1) Randomly pick two large prime numbers p and q, and construct n = p*q;
- (2) Calculate Euler's totient function φ(n) = (p-1) * (q-1);
- (3) Randomly pick e such that gcd(e, φ(n)) = 1, i.e., e and φ(n) are coprime; gcd refers to finding the greatest common divisor;
- (4) Calculate d such that e*d ≡ 1 (mod φ(n)), i.e., d is the multiplicative inverse of e.
2. Encryption process:
(1) The information to be encrypted (plaintext) is m, m < n; (because modular arithmetic is to be performed, if m is greater than n, the subsequent operations will not hold; therefore, when the information is larger than n, it should be encrypted in blocks);
(2)) The generation of ciphertext c is $$ c = m^e mod (n) $$
3. Decryption
$$ c^d mod (n) = (m^e)^d mod (n) = m^(d*e) mod (n) ; $$
3. Decryption
$$ c^d mod (n) = (m^e)^d mod (n) = m^(d*e) mod (n) ; $$
Why is decryption possible?
Euler's theorem is needed (actually a generalization of Fermat's little theorem)
a^φ(n) ≡ 1 (mod n)，
Further generalization: a^(φ(n)k) ≡ 1 (mod n)，
We get a^(φ(n)k+1) ≡ a (mod n)
Note that ed ≡ 1 mod φ(N), i.e., ed = 1 + k*φ(N)。
Therefore, $$ M^(de) mod N = M^1 + kφ(N) mod N = M
$$
4. The code is as follows
Example
#coding=utf-8
#__author__ = 'ralph'
import random
def extendedGCD(a, b):
#a*xi + b*yi = ri
if b == 0:
return (1, 0, a)
#a*x1 + b*y1 = a
x1 = 1
y1 = 0
#a*x2 + b*y2 = b
x2 = 0
y2 = 1
while b != 0:
q = a / b
#ri = r(i-2) % r(i-1)
r = a % b
a = b
b = r
#xi = x(i-2) - q*x(i-1)
x = x1 - q*x2
x1 = x2
x2 = x
#yi = y(i-2) - q*y(i-1)
y = y1 - q*y2
y1 = y2
y2 = y
return(x1, y1, a)
def computeD(fn, e):
(x, y, r) = extendedGCD(fn, e)
#y maybe < 0, so convert it
if y < 0:
return fn + y
return y
def keyGeneration(p,q,e):
#generate public key and private key
n = p * q
fn = (p-1) * (q-1)
d = computeD(fn, e)
return (d,n)
p_v = int(raw_input('Please enter the value of p (decimal)\n'))
q_v = int(raw_input('Please enter the value of q (decimal)\n'))
e_v = int(raw_input('Please enter the value of e (decimal)\n'))
c_v = int(raw_input('Please enter the value of ciphertext c (decimal)\n'))
(d,n) = keyGeneration(p_v,q_v,e_v) # Generate d and n
m = pow(c_v,d,n)
print ("The obtained plaintext m is:"+str(m))
When p value: 18443, q value: 49891, e value: 19,
Ciphertext c value:
70479679275221115227470416418414022368270835483295235263072905459788476483295235459788476663551792475206804459788476428313374475206804459788476425392137704796792458265677341524652483295235534149509425392137428313374425392137341524652458265677263072905483295235828509797341524652425392137475206804428313374483295235475206804459788476306220148
The result obtained will show
The obtained plaintext m is: 88455713
Example
#coding=utf-8
#__author__ = 'ralph'
import random
def extendedGCD(a, b):
#a*xi + b*yi = ri
if b == 0:
return (1, 0, a)
#a*x1 + b*y1 = a
x1 = 1
y1 = 0
#a*x2 + b*y2 = b
x2 = 0
y2 = 1
while b != 0:
q = a / b
#ri = r(i-2) % r(i-1)
r = a % b
a = b
b = r
#xi = x(i-2) - q*x(i-1)
x = x1 - q*x2
x1 = x2
x2 = x
#yi = y(i-2) - q*y(i-1)
y = y1 - q*y2
y1 = y2
y2 = y
return(x1, y1, a)
def computeD(fn, e):
(x, y, r) = extendedGCD(fn, e)
#y maybe < 0, so convert it
if y < 0:
return fn + y
return y
def keyGeneration(p,q,e):
#generate public key and private key
n = p * q
fn = (p-1) * (q-1)
d = computeD(fn, e)
return (d,n)
p_v = int(raw_input('Please enter the value of p (decimal)\n'))
q_v = int(raw_input('Please enter the value of q (decimal)\n'))
e_v = int(raw_input('Please enter the value of e (decimal)\n'))
c_v = int(raw_input('Please enter the value of ciphertext c (decimal)\n'))
(d,n) = keyGeneration(p_v,q_v,e_v) # Generate d and n
m = pow(c_v,d,n)
print ("The obtained plaintext m is:"+str(m))
The obtained plaintext m is: 88455713