Code and ciphers: Julius Caesar, the Enigma and the internet

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chapter 9
The ‘dots’ indicate places where we do not yet have enough information
to give the cipher letter. The full 26�26 encipherment table can be filled
in when we know the encipherment of all 26 letters at position 1.
Note the important feature of the encipherment table: each diagonal
from North-West to South-East constitutes a full alphabet, in normal
order, starting from the letter in column 1.
Alternatively, if we know the encipherment of A, or any other letter, at
all 26 positions of the wheel we can equally well work out the encipherment
of any letter at any position. For example, suppose that we wish to
know the encipherment of N at position 11. The wire that has N as its
entry point at position 11 is the wire that had A as its entry point 13 positions
earlier, since N is 13 places after A in the alphabet. Now
11�13��2, and position �2 is the same as position 26�2, i.e. 24.We
therefore look up what letter A enciphers to at position 24. If this is, say, G
then N at position 11 will encipher to the letter which is13 places after G in
the alphabet at position 11, that is, to T.
Readers who are familiar with matrices will recognise that what we are
doing, in effect, is representing the encipherment provided by a wheel as
a 26 �26 matrix. The first column gives the encipherment of the complete
alphabet at position 1 and the first row gives the encipherment of A at each
of the 26 positions.
The matrix can then be completely filled in from either its first row or
its first column by using the ‘diagonal property’ explained above. A cryptographic
feature of some importance is that whereas any column will
contain all 26 letters of the alphabet, since two letters cannot encipher to
the same letter at the same wheel position, the rows may contain one or
more letters twice or more, since there is nothing to prevent a letter enciphering
to the same letter at two or more positions. In fact, with a wheel
of size 26, or any even number of contacts, it is certain that each row will
contain at least one repeated letter. In the 6-letter example above this
already occurs, C goes to A at positions 1 and 3. With an odd number of
contact points it is possible that the rows will not contain any repeated
letters. From a cryptographic point of view the fewer repeated letters in a
row the better. (For an explanation of this, and further details see M14.)
Encipherment by the Enigma
We have just seen how a single wired wheel enciphers a letter. In the
Enigma the current from the keyboard letter passes through the entry