In the ElGamal example I linked you'll see that we use the same key for encryption and decryption. However, we perform a modification on the cyphertext using known values. This has a similar effect.

It's also somewhat definitional. We're taking an established encryption scheme in both cases and constructing a new algorithm on top of it. Above where I say we're performing decryption with a key set to key^2, we could also think of it as doing Dec(key, key*m) where the key is the same but we've done some operation on the cyphertext.

> In the ElGamal example I linked you'll see

Was on mobile phone so was too hard to read the handwritten text. Will definitely check it out later.

Anyway thanks again for the additional explanation, much appreciated.