The only hope left was that it might be possible to set messages by
using the statistical properties of the plain language, or extended psi-stream.
44B SETTING - STATISTICAL METHODS
(a) First ideas - P characteristics.
An attempt was made early in i942 to set X's and PSI's for a message by
using the observed fact that dots predominated markedly over crosses in the fifth
impulse of ordinary Tunny plain language. This was not successful but the
possibility of using this effect was again investigated. The chief difficulty
was the irregular movement of the PSI wheels, but it was hoped that the PSI' key
could be approximated to sufficiently closely by using a standard motor key
instead of the unknown true motor key. The theoretical investigation showed that
success might just be possible with ab not = 1/2 but that no success could be
expected with ab = 1/2. The reason for this was closely connected with the
predominance of changes in the PSI pattern: when the assumed setting of a PSI
wheel was one place off the true setting, the resulting sign in the assumed PSI'
key was more likely to be wrong than right.
(b) Delta-PSI' characteristics.
In another investigation, no attempt was made to use the periodicity of the
unextended PSI impulses but an attempt was made to derive a statistical method
from a consideration of the other non-random properties of the PSI' key. These
(i) All five PSI wheels have the same movement
and (ii) In the unextended PSI impulses, changes are much more
frequent than continuation.
These properties, it was thought, could best be expressed in terms not of
the actual PSI' key, but of its first difference, which we denote Delta-PSI' .
Changes and continuation in PSI' are represented by crosses and dots respectively
At this time, as in March and April, 1942, the Germans always arranged that
ab = 1/2 , so that dots and crosses were equally frequent in each impulse of
Delta-PSI'. Hence no statistical method could be founded, it was thought, on the
statistical properties of Delta-PSI'.
But suppose, it was argued, that two impulses of the Delta-PSI' key, say the
first and second, are added together. The resulting sequence Delta-PSI'1 + Delta-
PSI'2 will have a dot in each position corresponding to a dot in the motor key,
and in the positions corresponding to crosses in the motor key, the proportion of
dots will be (b)^2 + (1 - b )^2 , and the proportion of crosses 2b ( 1 - b ), if,
as an approximation we take the same value of b for each impulse. But then the
proportion of crosses in the entire sequence will be
2 ab ( 1 - b ) = 1 - b
and therefore the proportion of dots in the sum of any two impulses of Delta-PSI'
will be equal to b and about 70%.
It was deduced that the first step in any statistical method of wheel
setting should be the differencing of the cipher text and the addition of two
impulses of the resulting stream of letters.
All now depended an the properties of Delta-P1 + Delta-P2. Counts