Extended Group Autokey (XGAK)

Extended group autokey (XGAK) ciphers are an extension of group autokey (GAK) ciphers which, in terms of ciphertext alphabet permutations, are able to select different letters from the alphabet at each step depending on the plaintext alphabet letter that was used. (How do I explain this lol)

Like GAK ciphers, this kind of cipher consistently produces isomorphs for repeated plaintext sequences, so they are perfectly isomorphic. It's fairly easy to choose the plaintext mapping to ensure doubles will always be avoided. And like GAK, this kind of cipher produces chaining conflicts.

Based on these properties, XGAK is a good candidate cipher method for the eyes, however so is GAK, so based on Occam's razor we're choosing not to look too closely into XGAK at the moment, since it's just more complicated and there aren't currently any properties of the eyes that require XGAK vs. just GAK to be explained.

This is the last known cipher class in the isomorphic cipher hierarchy that is within the perfectly isomorphic region. It's currently unknown whether or not this covers all perfectly isomorphic ciphers.