12C Page 19
12C CHI BREAKING AND SETTING Solution of Z = Χ + D.
(a) Frequency of letters in ΔΨ'.
The precautions taken by the Germans in the construction of wheel patterns produce Χ, ΔΧ, and Ψ' streams in which each letter occurs an approximately equal number of times. But though the arrangements for ΔΨ' produce an even distribution of dots and crosses in each impulse separately, the fact that there is a dot on every impulse wherever there is an extension and a preponderance of crosses in other places, produces a ΔΨ' stream in which
wherever there is a TM dot there is a stroke
wherever there is a TM cross the frequency of the various letters in ΔΨ depend on the number of crosses in them.
It can easily be seen that the proportion of TM dots (which = 1-a) and the frequency of crosses in each impulse at TM cross positions (which = b) both increase with the dottage. Fig. 12 (II) gives a ΔΨ' count on a day with 26 dots in μ37.
(b) Frequency of letters in ΔP.
The number of occurrences of each letter in ΔP are by no means equal. The frequent repetition in P of groups of letters common in punctuation or German language like 55M889 or 5M89 (full stop) E1, EN N9, SCH and so on naturally implies the frequent repetition in ΔP of their differenced equivalents /UA/5, UA5, U, F, 3, JG. Therefore letters like 5 and U which come from popular bigrams in P are frequent in ΔP. Typical P and ΔP counts are given in Fig. 12 (II).
(c) Frequency of letters in ΔD.
We now consider what happens in ΔD = ΔP + ΔΨ'. Wherever ΔΨ' is a stroke, ΔP will be reproduced in ΔD, since any letter added to stroke is unchanged. Therefore the shape of the ΔD count at those places where there are TM dots is identical with the shape of the count of ΔP. In other places every letter in ΔD will occur an approximately equal number of times (though the combination of letters common in ΔP and in ΔΨ' against BM crosses will make some letters rather stronger than the others). A ΔD count can therefore be regarded as a watered down version of the ΔP count at the back of it.
As the μ37 dottage increases and therefore also the proportion of strokes in ΔΨ', the proportion of ΔD count derived directly from ΔP count increases, and the ΔD count from a given ΔP count will be correspondingly stronger.
(d) Chi Setting.
It has been shown that ΔD = not only ΔP + ΔΨ' but also ΔZ + ΔΧ. If we know the wheel-patterns, we can (in theory) set the chi wheels to every possible combination of settings in turn and generate all the ΔΧ sequences corresponding to the Χ streams with which the transmission could have been enciphered. These can be added to the ΔZ and all possible ΔD's obtained. The counts of letter frequency in all these possible dechis will be more or less level, except the counts of the correct dechi which will follow the pattern described in the last paragraph. If the correct count shows the characteristics of a ΔD count strongly, it will be easily identified, and the chi settings will be found without any doubt