transmission had now been superseded by tone transmission in 5 - unit code; near
depths were once more appearing. Many of these were corrupt, but the beginnings
of some were decoded, and were shown to be of the same stereotyped forms as were
those of July and August 1941. Two or three hundred letters of one February depth
were read and an attempt was made to break the machine. This failed. A near
depth of March 3rd was passed over in favour of the February depth.
(b) A depth of three
On March 29th, an unprecedented phenomenon, the interception of a depth of
three, occurred. Attention was immediately diverted to it. Reading in depth of
three was found to be very easy, and it was soon carried to the end of the
shortest of the three messages (975 letters). It was continued for the other two
messages without a break up to the 1060th letter. There was no ambiguity about
the subtractor key, as there would have been in a depth of two, and there was
hardly any possibility of corruption in it, since all three messages were good,
and since two messages would need to be corrupt in the same letter in order to
produce an error in the calculated subtractor key. No better length of key could
have been desired, and all the energies of the Research Section were thrown into
the attempt to break it, but without success. Some evidence was found to confirm
the hypothesis that the periods of the X wheels were the same as of old, but that
was all. It was supposed that the Germans had taken steps to eliminate the non-
random characteristics of the extended PSI patterns. The Research Section did
not manage to anticipate Turing's Method of Key Analysis and work on the depth
of three had to be abandoned.
(c) A near depth of March 3rd.
However, though depths could no longer be broken, it was thought that a near
depth might prove vulnerable. For when a near depth can be read it gives not
merely one key, but two closely related, but different keys. Attention was
therefore transferred to the rather corrupt near depth of March 3rd. which has
already been mentioned.
The two messages of the near depth had indicators which differed only in two
of the last five letters, and therefore according to the hypothesis referred to in
section IV the only difference between the two subtractor keys was in the settings
of two of the X wheels.
The near depth was decoded for about 30 letters and the sum Ka + Kb of the
two keys was determined. Crosses (of course) appeared only in the impulses whose
X wheels had different settings in the two messages. Both these impulses of Ka +
Kb should have shown the periodicity of the corresponding X wheels, and were in
fact found to do so, though the piece of pattern actually repeated in either
impulse was of course very short. Hence, both these impulses were assumed to be X
patterns "differenced" at some unknown interval. By repeating the patterns the
sequence Ka + Kb could be extended as far as was desired. So from this sequence
and the cipher texts the sum of the two clear texts could be derived. This sum
was attacked as in the breaking of ordinary depths, and two or three hundred
letters were decoded. So two alternatives for the subtractor key of either of the
messages were worked out for this stretch of two or three hundred letters.
This success established the validity of the assumptions which led up to it.
At this stage then, not only were two alternatives for a length of key known
but also two X patterns differenced at unknown interval had been obtained.