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I understand how fractionation is used in the straddling keyboard, which is an important feature of the VIC Cipher, and in the Polybius Square of the ADFGVX cipher, but I am not sure how to describe its cryptographic function and benefit.

It helps spread the redundancies of the plaintext over the ciphertext, and that is diffusion. In doing so, it increases a message's uncertainty.

But on the face of it, fractionation also looks like a kind of substitution.

What exactly is fractionation, and what can it achieve in terms of information theory?

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...looks like a kind of substitution.

It's perhaps more similar to encoding, say to ASCII, Baudot code or "dits & dahs" (morse code). And as the name implies, fractionation breaks up the message characters into smaller pieces (fractions) that then exactly represent those characters.

It was used in the more classical ciphers like transpositional ones. Probably before the common use of computing machines. The transpositions occurred on a character by character basis. The coordinates on a Polybius square can serve a similar function to that of an index into an S-box/lookup table. Or they can be otherwise manipulated in ways that a single character can't be. Since the advent of modern computers, this fractionation occurs automatically when the characters are represented as bits within a block of bytes. The term does not seem to appear in contemporary cryptography.

I suggest that the benefit is not for elimination of message redundancy, but for assisting with diffusion. Substitution and permutation are entirely mathematically possible with a human alphabet. You'd just have multi-character substitution boxes, so a three character substitution box would accept input like ggn and output yqy. You'd permute single letters between them as we do today. It would be cumbersome though as such an s-box would be at least 19,683 entries long ($(26 + 1)^ 3$). Numbers extra. And their efficient generation via polynomials would be problematic. Algebraic qualities, avalanche and linearity were not well appreciated then.

So now we fractionate/encode to numbers and bits, and manipulate, substitute and permute them instead.

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  • $\begingroup$ @MaartenBodewes Err, it's "dit, dit da, dit". And their plural. That's what the Brits kids use for Morse. It'd have thought that was common urban slang. I must be too cool for my own good :-) It's also a (very) thinly veiled reference to the wiki article on frationation that uses Morse as an example encoding. $\endgroup$
    – Paul Uszak
    Commented Sep 22, 2019 at 12:14
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    $\begingroup$ This is an international audience Paul, if you want to give a reference to a Wiki article then it probably best starts with .... - - .--. ... .-.-.- It seems that everybody else spells this as "dits and dahs" by the way. Note that I had to spell check your otherwise fine answer as well, including the word "fractionation" which is what this post is about. $\endgroup$
    – Maarten Bodewes ♦
    Commented Sep 22, 2019 at 13:47
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    $\begingroup$ Morse operators and collectors often say "dits and das". The words are onomatopoeic. $\endgroup$
    – Patriot
    Commented Sep 22, 2019 at 14:35

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