Isomorphs (Gap Patterns)

Isomorphs are repeating patterns of repeated ciphertext symbols, where the symbols used can be different each time, but the relative positions of repeats are the same, so that all of the occurrences are substitutions of each other.

(I really need to think of a better one-sentence explanation that is accurate without being excessively long or confusing...)

Isomorphs can be identified by their gap patterns, showing the positions of repeated symbols within a segment of ciphertext. The format used for gap patterns is:

• . is used to represent non-repeated symbols. Note that every occurrence of . should be a distinct symbol.
• Letters A, B, C, ... are used to represent repeated symbols, in the order they occur. The first repeated symbol is replaced with A, the next repeated symbol which is different from A is replaced with B, etc.

For example, in the eyes, the most prominent isomorph which was first noticed was the following set of 5 sequences occurring in the first 3 messages (written in ASCII+32, though the form the symbols take does not matter, only which ones are the same, and writing it in a format where one trigram is one character makes it easier to see), with one of them occurring twice:

OLPJ3P-O3 (twice)
g+jX$j3g$
dN1D-15d-
&-`=Q`_&Q
IhY47YaI7

All of these sequences have the same gap pattern:

A.B.CB.AC

Additionally, all of these occur in similar places in the messages, with two per message. The exact positions they occur are:

• East 1: First at trigram 41, distance 28 between them
• West 1: First at trigram 41 (same position and CT as East 1), distance 30 between them
• East 2: First at trigram 46, distance 35 between them

Due to this identical pattern and similarity in placement within these messages, it was conjectured that these could be due to the same plaintext word or phrase being repeated in these places. Then we found that there are many kinds of ciphers which produce these isomorphs, and they are produced exactly when a word or phrase is repeated in different cipher states, where the state changes in the same way across the course of the word/phrase.

Many other isomorphs exist in the eyes, and many of them are strong patterns with multiple repeated symbols with similar placement in multiple messages like this, so there is very high confidence that the eyes are a cipher which produces isomorphs and these isomorphs come from repeated segments of plaintext.

It's important to note that the gap pattern of an isomorph is not the same as the gap pattern of the plaintext word/phrase used to produce it. Instead, ciphers which produce isomorphs generally produce the same gap pattern consistently for any particular word, but the positions of the repeated symbols in that pattern are not directly related to the repeated symbols in the plaintext at all. The exact details of the cipher mechanism lead to any given word forming a particular gap pattern in an unpredictable way, and not all words produce strong gap patterns with multiple repeats either.

It's possible to have isomorphs from repeated plaintext which don't end up being readily apparent in the ciphertext, because they have fewer repeated symbols, or none at all. It's only the most clearly visible isomorphs that we're able to see and work with.

The isomorphs are a major piece of the puzzle that is the eyes. They tell us that the cipher method is something which is capable of producing isomorphs, which allows us to say the eyes must be a polyalphabetic substitution cipher, and at the same time they show us directly some of the places that plaintext is being repeated, showing that the messages actually have quite a lot of repeating content, especially in sets of 3 messages (even more than the more immediately noticeable shared sections).

One caveat with gap patterns is that even in ciphers that produce isomorphs, it's possible to have the same gap pattern occur despite the plaintext being different, just from having two different words happen to produce the same pattern. This becomes much less likely the more repeated symbols the isomorph has in it, and seeing multiple occurrences of a pattern at similar positions in multiple messages also helps to confirm that it's not a coincidental repeat, but it is still possible for the same gap pattern to occur for different words by chance.