Commutative Encryption using a Modified ElGamal Method

My first real interaction with cryptography was a PhD related to the usage of commutative encryption:

It used a derivative of the RSA method, and which is known as SRA. But, this method struggles with its security (as we need to keep the modulus secret).

And, so, this week the mighty Tahir ElGamal is coming to give a guest talk to our students, so let’s implement commutative encryption using the ElGamal method.

Commutative encryption with a modified ElGamal method

So, for this, I will use this paper [1][here]:

With commutative encryption, we can apply Bob’s public key to a message, then encrypt with Alice’s public key. We can then decrypt with Bob’s private key, and then Alice’s private key, or decrypt with Alice’s private key and then Bob’s public key. This differs from the normal encryption process, and where we would decrypt using the reverse of the application of the keys.