ElGamal Encryption Algorithm

ElGamal encryption is an public-key cryptosystem. It uses asymmetric key encryption for communicating between two parties and encrypting the message.
This cryptosystem is based on the difficulty of finding discrete logarithm in a cyclic group that is even if we know ga and gk, it is extremely difficult to compute gak.

Idea of ElGamal cryptosystem
Suppose Alice wants to communicate to Bob.

  1. Bob generates public and private key :
    • Bob chooses a very large number q and a cyclic group Fq.
    • From the cyclic group Fq, he choose any element g and
      an element a such that gcd(a, q) = 1.
    • Then he computes h = ga.
    • Bob publishes F, h = ga, q and g as his public key and retains a as private key.
  2. Alice encrypts data using Bob’s public key :
    • Alice selects an element k from cyclic group F
      such that gcd(k, q) = 1.
    • Then she computes p = gk and s = hk = gak.
    • She multiples s with M.
    • Then she sends (p, M*s) = (gk, M*s).
  3. Bob decrypts the message :
    • Bob calculates s = pa = gak.
    • He divides M*s by s to obtain M as s = s.

Following is the implementation of ElGamal cryptosystem in Python

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# Python program to illustrate ElGamal encryption
  
import random 
from math import pow
  
a = random.randint(2, 10)
  
def gcd(a, b):
    if a < b:
        return gcd(b, a)
    elif a % b == 0:
        return b;
    else:
        return gcd(b, a % b)
  
# Generating large random numbers
def gen_key(q):
  
    key = random.randint(pow(10, 20), q)
    while gcd(q, key) != 1:
        key = random.randint(pow(10, 20), q)
  
    return key
  
# Modular exponentiation
def power(a, b, c):
    x = 1
    y = a
  
    while b > 0:
        if b % 2 == 0:
            x = (x * y) % c;
        y = (y * y) % c
        b = int(b / 2)
  
    return x % c
  
# Asymmetric encryption
def encrypt(msg, q, h, g):
  
    en_msg = []
  
    k = gen_key(q)# Private key for sender
    s = power(h, k, q)
    p = power(g, k, q)
      
    for i in range(0, len(msg)):
        en_msg.append(msg[i])
  
    print("g^k used : ", p)
    print("g^ak used : ", s)
    for i in range(0, len(en_msg)):
        en_msg[i] = s * ord(en_msg[i])
  
    return en_msg, p
  
def decrypt(en_msg, p, key, q):
  
    dr_msg = []
    h = power(p, key, q)
    for i in range(0, len(en_msg)):
        dr_msg.append(chr(int(en_msg[i]/h)))
          
    return dr_msg
  
# Driver code
def main():
  
    msg = 'encryption'
    print("Original Message :", msg)
  
    q = random.randint(pow(10, 20), pow(10, 50))
    g = random.randint(2, q)
  
    key = gen_key(q)# Private key for receiver
    h = power(g, key, q)
    print("g used : ", g)
    print("g^a used : ", h)
  
    en_msg, p = encrypt(msg, q, h, g)
    dr_msg = decrypt(en_msg, p, key, q)
    dmsg = ''.join(dr_msg)
    print("Decrypted Message :", dmsg);
  
  
if __name__ == '__main__':
    main()

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Sample Output :

Original Message : encryption
g used :  5860696954522417707188952371547944035333315907890
g^a used :  4711309755639364289552454834506215144653958055252
g^k used :  12475188089503227615789015740709091911412567126782
g^ak used :  39448787632167136161153337226654906357756740068295
Decrypted Message : encryption

In this cryptosystem, original message M is masked by multiplying gak to it. To remove the mask, a clue is given in form of gk. Unless someone knows a, he will not be able to retrieve M. This is because of finding discrete log in an cyclic group is difficult and simplying knowing ga and gk is not good enough to compute gak.



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