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

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chapter 6
before, to remind ourselves how it is done, let us look at an example.
Suppose that we have the code group 6394 and that the key to be applied
to it is 2798; then the code group is written down, the key placed directly
underneath it and corresponding digits are added without carrying so that
when we add the last digits of the code and key, 4 and 8, the sum is written
as 2, not 12 (that is: we are adding digit by digit (mod 10)). So we have
Code group 6394
Key 2798
Sum 8082
and the cipher text is 8082. The key would not be the same for the other
groups since in practice the key either would not repeat at all or, if it did
repeat, would only do so after many digits. Since encryption involves adding
the key to the code groups the person receiving the message would have
to subtract the key digit by digit (mod 10) from the cipher in order to
recover the code groups and so decipher the message; thus:
Cipher 8082
Key 2798
Code group 6394
Obviously the code groups are now disguised, and the security of the
system is substantially increased provided that the key does not repeat for
a sufficiently long period. The question of how to produce sequences of
digits which do not repeat until many thousands have been generated is
one of considerable interest to mathematicians and cryptographers, and
we consider it more fully in Chapter 8. In the mean time, by way of illustration,
here is a very simple method, which generates a sequence which
repeats after 60 digits.
Example 6.2
Generate a sequence of digits (mod 10) by starting with the digits 3 and 7
and forming each new digit by adding together the two previous digits
(mod 10).
Solution
The sequence starts
3 7
so the next digit is (3�7)�10 which is 0 (mod 10) and the 4th digit is
(7�0)�7 and so the 5th digit is (0�7)�7. Continuing in this way we find
that the sequence which is generated is