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

and so
2�4�6�12
�(4095)�4095� �1728
3�5�7�13
and the number of binary linear recurrences of order 12 which generate
binary sequences of maximum length 4095 is therefore
�144
as stated in Chapter 8. Note that although 4095 is divisible by 3 2 the fraction
in the Euler function is two-thirds, not eight-ninths.
Similarly, when k�23 the number of binary linear recurrences of order
23 which generate binary sequences of maximal length (2 23 �1) is
which is
1728
12
�(2 23 � 1)
23
�(47�178 481)
23
and 47 and 178 481 are primes (that 178 481 is a prime follows from the
fact that it is not divisible by any prime less than its square root, i.e. by
any prime less than 422). We therefore find that the number of such
sequences is
46�178 480
23
�356 960
which was also mentioned in Chapter 8.
(For a proof of the theorem quoted above see [8.2].)
M12 Recovery of a binary linear recurrence from a
stretch of key
Example
The following 15 binary digits of key have been recovered from a cipher
message:
10101 00110 00100
Mathematical aspects 199
the most recent digits being to the right. Verify that these can be generated
by a linear recurrence of order 5.