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 kn

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.