Question

What’s an elgamal cryptography ? And how does it work?

Answer 1

ElGamal cryptographic system is based on asymmetric key encryption. It is a public-key cryptosystem. This system ensured an extra layer of security as it depends on the difficulty of finding discrete logarithm in cyclic group. It was invented in 1985 by Egyptian cryptographer Taher Elgamal. It uses a public and private key pair for encryption and decryption.

AlGamal encyption algorithm: Let’s say the two communicating parties are Alice (sender) and Bob (recipient).

The public key and the private key generation by Bob:

• First a very large number (q) and a cyclic group (F_q) needs to be chosen.
• Then two elements g and a needs to be chosen from that cyclic group (F_q), such that gcd of g and a is 1.
• Then g^a = h.
• Now the cyclic group F, generated h, and g and q needs to be published as public key, and a needs to be retained as private key.

Encryption by Alice using public key generated by Bob:

• An element k is to be selected from the cyclic group F, such that the gcd of q and k is equal to 1.
• then s and p are computed, such that s = h^k = g^ak and p = g^k
• s and message, M are multiplied.
• Then the encryption is sent as (p, M*s) = (gk, M*s).

Decryption of the message by Bob:

• s′ is calculated such that s′ = pa = gak
• Then the message is decrypted by dividing M*s by s′, as s = s′.

Why this works?

This because both public and private key are taken into account while decryption. Here the message is morphed by multiplying it with g^ak. In order to decrypt the message, g^k is provided as clue. Without having the private key a, it is impossible to retrieve the message M. Even if g^k and g^a are known, finding g^ak is extremely difficult as it involves finding discrete log in a cyclic group.

Screenshot of the program implemented in python: