ElGamal Encryption Algorithm

ElGamal Encryption is a public-key cryptosystem. It uses asymmetric key encryption to communicate between two parties and encrypt the message. This cryptosystem is based on the difficulty of finding discrete logarithms in a cyclic group that is even if we know g^a and g^k, it is extremely difficult to compute g^ak. In this article, we will learn about the Elgamal algorithm, the components of its algorithm, its advantages & disadvantages, and the implementation of the ElGamal cryptosystem in Python.

Elgamal Cryptographic Algorithm

The ElGamal cryptographic algorithm is an asymmetric key encryption scheme based on the Diffie-Hellman key exchange. It was invented by Taher ElGamal in 1985. The algorithm is widely used for secure data transmission and has digital signatures and encryption applications. Here’s an overview of its components and how it works:

Components of the ElGamal Algorithm

1. Key Generation:
   • Public Parameters: Select a large prime number p and a generator g of the multiplicative group Z^*_p.
   • Private Key: Select a private key x such that 1 ≤ x ≤p −2.
   • Public Key: Compute h=g^x mod  p. The public key is (p,g,h) and the private key is x.
2. Encryption:
   • To encrypt a message M:
     • Choose a random integer k such that 1 ≤ k ≤ p−2.
     • Compute C₁ = g^k mod  p.
     • Compute C₂ =M⋅h^k mod  p.
     • The ciphertext is (c1,c2).
3. Decryption:
   • To decrypt the ciphertext (c1,c2) using the private key x:
     • Compute the shared secret s= C^x₁ mod  p.
     • Compute s⁻¹ mod  p (the modular inverse of s).
     • Compute the original message M = C₂⋅s⁻¹ mod  p.

Idea of ElGamal Cryptosystem

Suppose Alice wants to communicate with Bob.  

1. Bob generates public and private keys: 
   
   • Bob chooses a very large number q and a cyclic group F_q.
   • From the cyclic group F_q, he choose any element g and
     an element a such that gcd(a, q) = 1.
   • Then he computes h = g^a.
   • Bob publishes F, h = g^a, q, and g as his public key and retains an as a 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 = g^k and s = h^k = g^ak.
   • She multiples s with M.
   • Then she sends (p, M*s) = (g^k, M*s).
3. Bob decrypts the message : 
   
   • Bob calculates s^′ = p^a = g^ak.
   • He divides M*s by s^′ to obtain M as s = s^′.

Following is the implementation of the ElGamal cryptosystem in Python 

# 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() 

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, the original message M is masked by multiplying g^ak to it. To remove the mask, a clue is given in form of g^k. Unless someone knows a, he will not be able to retrieve M. This is because finding discrete log in a cyclic group is difficult and simplifying knowing g^a and g^k is not good enough to compute g^ak.

Applications of ElGamal Encryption Algorithm

1. Encryption: ElGamal is used for encrypting messages where public key cryptography is required.
2. Digital Signatures: A variant of ElGamal is used for creating digital signatures, ensuring message authenticity and integrity.

Advantages

• Security: ElGamal is based on the discrete logarithm problem, which is considered to be a hard problem to solve. This makes it secure against attacks from hackers.
• Key distribution: The encryption and decryption keys are different, making it easier to distribute keys securely. This allows for secure communication between multiple parties.
• Digital signatures: ElGamal can also be used for digital signatures, which allows for secure authentication of messages.

Disadvantages

• Slow processing: ElGamal is slower compared to other encryption algorithms, especially when used with long keys. This can make it impractical for certain applications that require fast processing speeds.
• Key size: ElGamal requires larger key sizes to achieve the same level of security as other algorithms. This can make it more difficult to use in some applications.
• Vulnerability to certain attacks: ElGamal is vulnerable to attacks based on the discrete logarithm problem, such as the index calculus algorithm. This can reduce the security of the algorithm in certain situations.

Conclusion

In summary, the ElGamal encryption algorithm is a key method in modern cryptography. It provides a secure way to send data and create digital signatures using public and private keys. Its security is based on complex math problems, making it reliable. ElGamal is versatile, working in many different areas of cryptography. However, with the rise of quantum computing, we must keep updating and adapting our cryptographic methods to stay secure.