416 | 38 BANBURISmUS REVISITED
Where the same letter appears twice or more in a column (as N in position 182 or A in
position 185) this is called a ‘repeat’. A two-letter sequence is called a bigram. In Chapter 13 it
was explained how a repeating bigram (e.g. KS in columns 190–191) provides more powerful
evidence than two single repeats. Repeating trigrams or better are more powerful still.
The cryptanalysts, unless they had the good fortune that the indicator’s enciphering system
had already been solved, needed a different way of getting started. Applying Banburismus to the
numbers of repeats, in order to set messages in depth, provided one way.
The scale of the problem
It is important first to get a grasp of the number of variations of which Enigma was capable, and
the consequent complexity of solving it anew each day. There were of course variations between
one Enigma model and another, and between different periods of their use. The following fig-
ures are typical for the three-wheel Naval Enigma machine with ten leads on the plugboard.
The three wheels in order can be chosen from the available eight in 8 × 7 × 6 = 336 ways.
Their three ring settings can be chosen in 26^3 = 17,576 ways. Combining these gives 5.91 × 10^6
combinations from the wheels alone. On the plugboard, for the first lead, there are 26 ways
of inserting the first end and 25 of inserting the second. For the second lead, there are 24 and
23, respectively. Then 22 and 21, and so on down to 8 and 7 for the tenth lead. These multiply
together to give 5.6 × 10^23 combinations. But we have overcounted. The order of choosing the
ten leads is irrelevant, so to avoid multiple counting we have to divide by 10 × 9 × 8 × 7 × 6 × 5 ×
4 × 3 × 2 = 3.63 × 10^6 , and since it is also irrelevant which end of a lead is inserted first, we have
to divide again by 2^10 = 1024. The combined divisor, 3.72 × 10^9 , reduces the number of different
plugboard combinations to 1.51 × 10^14. When these are combined with the 5.91 × 10^6 wheel
orders and ring settings, we obtain the final total of 8.9 × 10^20 , or 890 million million million
daily keys. This is some 75 million million times the number for a five-letter combination safe
(see Chapter 13).
The often-quoted smaller figure of 159 million million million keys relates to the Army or
Air Force Enigma which had three wheels chosen from five, instead of the Naval Enigma’s
choice of three from eight (Banburismus was not used against Army or Air Force Enigma).^1
Tetras and the Hollerith section
As we saw in Chapter 13, pairs of messages with only one indicating letter in common were
so numerous that nothing less than a repeat of four consecutive letters (a tetragram) was
Table 38.1
Position
in table180 181 182 183 184 185 186 187 188 189 190 191 192 193Message 1 Q V A J X P B K S B
Message 2 W R N D M F C I R U K S D Y
Message 3 N F Y A M S A