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

combining the keys of two or more linear recurrences, as the following
simple example shows.
Example 8.2
Use the (mod 2) sum of the keys generated by the two linear recurrences
and
U n �U (n�1) �U (n�2) , U 0 �U 1 �1
U n �U (n�1) �U (n�3) , U 0 �U 1 �U 2 �1
to produce a new key. Verify that this has a period of 21.
Verification
The first recurrence, as we have seen, has a period of length 3 and produces
the keystream
110110110110...
The second recurrence has a period of length 7 and produces the key
stream
111010011101001110100....
Producing random numbers and letters 105
Writing both of these out one under the other and adding the bits (mod 2)
we have
1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0.... .
1 1 1 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0.... .
Adding (mod 2) 0 0 1 1 0 0 1 0 1 0 1 1 1 1 1 0 0 0 0 1 0 0 0 1 1 0 0....
.
and we see that the key repeats after 21 places, but not before. Since the
first key has period 3 and the second key has period 7 the period of the
combined key cannot exceed 21, for both keys repeat after 21 places. On
the other hand, since 3 and 7 have no common factor, the combined key
cannot repeat after less than 21 places.
There is no need to restrict ourselves to the use of two linear recurrences;
we could use three or more. The advantage would be that the more we use
the more difficult it would be for a cryptanalyst to solve the system. The
disadvantage, if we are working by hand, would be the tedious nature of
the key generation and the increased probability of errors. Of course if we
have a means of generating the key by either a mechanical or an electronic
device the disadvantage disappears. It is not, therefore, surprising that