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

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chapter 8
multiplier built into the program, and the choice of seed left to the user. To
ensure even better results more than one such generator may be used and
their outputs combined in some way. Further improvement can be
achieved if the numbers are not used in the order in which they are produced,
but some form of ‘shuffling’ is employed to reduce the risk of correlation
between consecutive numbers. In this way good, long,
‘pseudo-random’ sequences may be generated. Among the known ‘good’
choices for A, B and M are those shown in Table 8.1.
Table 8.1
A B M
106 11 283 116 075
171 11 213 153 125
141 28 411 134 456
421 54 773 259 200
At the other extreme, a single generator with badly chosen values of A,
B and M may produce key with a very short period. Here is a small scale
example to illustrate these situations.
Example 8.4
Use the recurrence
U n �3U (n�1) �4 (mod 17)
to generate a sequence of integers starting with (1) U 0 �5, (2) U 0 �15.
Generation
(1) Since U 0 �5, U 1 �3�5�4�19�2 (mod 17) etc. The 16-long sequence
is
5, 2, 10, 0, 4, 16, 1, 7, 8, 11, 3, 13, 9, 14, 12, 6, 5, 2, 10,
... .
This generator, though modest, produces the maximum possible cycle,
which is 16 in this case. For additional comments see M13.
(2) U 0 �15 gives U 1 �3�15�4�49�15 (mod 17), so the period is
1! This explains why the value 15 doesn’t occur in the 16-long cycle
above.
Problem 8.2
Verify that the mid-square method which uses four-digit numbers starting
with X�7789 degenerates into a sequence of four numbers.