is no doubt that it is only a psychological one, and is not necessitated by the
nature of the machine and of the indicator system. The first effect suggested
that the last letter of the indicator controlled the setting of a wheel of
(b) Chis, Psis and extensions.
The first success in the analysis or the key was obtained towards the end of
January 1942 when it was found almost accidentally that many repeats occurred in
the first impulse of the key at intervals which were multiples of 41. This
suggested that this first impulse was the sum or a periodic sequence (of period
41) and of an aperiodic but non-random sequence. We here denote the periodic
sequence by X and the non-periodic sequence by PSI.
In order to reconstruct the sequences X and PSI , the first impulse was
written out on a width of 41, and for each set of five consecutive columns a count
was made of the five consecutive characters which occupied these columns. When
two such counts were made it was found that they were closely related; by adding a
constant set of five consecutive characters to each of the five character
sequences in one of the sets of columns, the frequency count of this set could be
brought into close agreement with that of the other. It was found that these
constant five-character sequences could be so chosen as not only to bring all the
frequency counts into good agreement, but also to fit together in their proper
order to form a periodic sequence of period 41. This sequence was denoted by X
and the result of adding it to the first impulse was denoted by PSI .
When PSI was examined with the object of determining its non-random
properties, the following "local" peculiarities were observed: -
(1) Consecutive signs in the sequence PSI tended to be equal.
In fact there was equality in about 3/4 of the cases.
(ii) The sequences .x. and x.x were significantly rare in PSI ,
even when the result (i) was taken into account.
It was then seen that the X pattern could have been reconstructed by
considering only pairs of consecutive columns in the rectangle, and that the power
of the method was not appreciably increased by taking five columns rather than
three. When the method came to be applied to other depths, the counts were
therefore made on sets of three consecutive columns.
The most striking property of PSI, was that it was roughly periodic; it
could be regarded as a periodic sequence of period 43 which had been "extended" by
replacing some dots by sequences of two or more consecutive dots, and some crosses
by sequences of two or more consecutive crosses. The PSI sequence was evidently
generated by a wheel of period 43 which sometimes moved on one place, and
sometimes stayed still when the cipher machine moved from one of its states to the
We may here introduce a slight change of notation. The extended key which
has been called PSI is now denoted by PSI' and the symbol PSI is used for the
periodic sequence from which it is derived by extension.
We have now reached the stage at which the first impulse was shown to be the
sum of a periodic sequence X of period 41, and an "extended" sequence PSI' derived
from a periodic sequence PSI of period 43. An ambiguity arose here, for the
pattern of X and PSI could both be reversed (by replacing dots by crosses, and
crosses by dots) without affecting their sum, but this was evidently of very