12A Page 16
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12 - CRYPTOGRAPHIC ASPECTS.
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12A THE PROBLEM.
(a) Formulae and Notation.
In Chapter 11 we have defined P, K, Χ, Ψ', and Z as the letter of plain language, key, chi, extended psi and cipher streams in the active position, P and so on as their predecessors, P and so on as their successors and ΔP = P + P etc.
Before discussing the cryptographic aspects of the Tunny machine it is necessary to restate the formula of the machine.
| Z = | P | + | K |
| K = | Χ | + | Ψ' |
and to list the following relevant variants, D (or DE-CHI) being defined as the sum of Z and Χ streams.
| Z | = | P | + | K | = | D | + | Χ | |
| K | = | P | + | Z | = | Χ | + | Ψ' | |
| D | = | Z | + | Χ | = | P | + | Ψ' |
For practical, if not logical, simplicity it will be found that P, K, Ψ', D and Z are sometimes used to refer not to any specific letter in the active position but to the whole of the stream concerned.
Further, now that the distinction between a message and a transmission has been carefully drawn, it will be convenient to refer to each of these as a message. This practice is in accordance with the traditional usage and agrees with that found in the Research Logs and other contemporary Tunny documents. The exact meaning will usually be clear from the context.
(b) Wheel-breaking and Setting.
Cryptographic work on Tunny fall into two parts
(i) The recovery of wheel patterns or WHEEL-BREAKING
(ii) The recovery of message settings or SETTING.
The theoretical basis of wheel-breaking and setting is very similar, and for every method of setting there is a corresponding method of wheel-breaking which uses more traffic and more information.
Normal practice is therefore to select the most promising material enciphered on a given set of wheel patterns and to use this for wheel-breaking. When the wheel patterns are known, they can then be used for setting other messages enciphered on them.
It will be noticed that it is possible to determine
(i) relative but not absolute settings
(ii) wheel patterns of corresponding chis and psis (e.g. Χ4 Ψ4) only with the proviso that dots and crosses may be interchanged on both wheels. This does not apply if one of the wheels is involved in the limitation.
(c) Weaknesses of Tunny.
The fact that Tunny can be broken at all depends on the fact that P, Χ, Ψ', K and D have marked statistical, periodic or linguistic characteristics which distinguish them from random sequences of letters.
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