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

What about sequences of higher order? A sequence of order 12, for
example, might have a period as long as 2 12 �1, which is 4095. Does any
linear recurrence of order 12 generate such a maximal binary sequence ?
As a result of some elegant and advanced mathematics a formula is
known which tells us exactly how many binary linear recurrences of order
k will produce a sequence of maximum period. In the case of recurrences
of order 12 the formula tells us that 144 binary linear recurrences of order
12 will produce binary sequences of period 4095. (The fact that 144�
12 � 12 is a fluke!) The remaining 3952 will fail to do so. The mathematical
analysis does not lead directly to these 144, which must be found by a
process somewhat akin to finding prime numbers. Alternatively, using a
computer, the 4095 possible recurrences can be tested and any which
repeat before 4095 terms have been generated can be rejected. When this
is done the first successful sequence found is
U n �U (n�6) �U (n�8) �U (n�11) �U (n�12) .
It is the ‘first’ in the sense that, writing 0 and 1 for the multipliers of the
12 terms on the right-hand side of the linear recurrence, this sequence has
the 12-bit representation
000001010011
which, interpreted as an integer written in binary form, is
64�16�2�1�83.
Producing random numbers and letters 103
No sequence with such an integer representation below 83 produces the
maximum period of 4095. For some of the mathematics behind all this
see M11.
By choosing a sufficiently high order and finding a linear recurrence
that gives the maximum period we can produce a key stream that would
seem to provide a pseudo-random binary stream which we could use as a
key for encryption. It can be shown, for example, that 356 960 linear
recurrences of order 23 will generate maximal key streams, which are
more than 8 000 000 long (M11). Since the initial starting values also
provide over 8 000 000 possibilities one might think that such a key
would present a formidable problem to the cryptanalyst and, initially, it
would. Unfortunately for the cryptographers key which has been generated
in this way has a fatal flaw: given a fairly modest amount of the key
the linear recurrence by which it has been generated and the initial values
can be recovered This is a consequence of the following.