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

142
chapter 10
(a) 0, 2, 3, 4, 8, 10;
(b) 1, 2, 2, 3, 4, 15;
(c) 1, 2, 2, 4, 5, 13;
(d) 2, 3, 3, 3, 4, 12;
(e) 2, 3, 3, 3, 5, 11;
(f) 2, 3, 4, 4, 7, 7.
The theoretical ‘work factor’ for the Hagelin
When a cryptanalyst is faced with messages on a totally unknown Hagelin
how many possibilities are there? This number is sometimes referred to as
the work factor and represents the number of cases that would have to be
tried if the cryptanalyst attempted a ‘brute force’ attack. For the type of
Hagelin that we have been discussing so far (there are other optional features
which complicate things further, as we shall see) there are two relevant
factors:
(1) the number of possible cages;
(2) the number of possible pin settings.
Assuming that the cage uses 27 lugs, and that we allow wheels to have 0
lugs, there are over 200 000 ways of distributing them between the six
wheels, so we’ll take the first factor as 2�10 5 . The second factor can be
written down at once. There are 131 pins and each pin can be in either
of two positions which means that there are 2 131 possible pin settings.
Since 2 131 is more than 2.5�10 39 the product of the two factors
exceeds
5�10 44 .
Because the cryptanalyst can’t be sure that there are 27 lugs being used he
would have to be prepared to try other cages, such as those having only 26
or 25, which would approximately treble the work factor. It is clear from
these numbers that a ‘brute force’ attack is out of the question. In fact
cryptanalysts do not have to resort to ‘brute force’ attacks to solve the
Hagelin, although solving the machine from cipher messages alone is difficult
and requires a lot of text.
The situation is quite different if the cryptanalyst has managed to
acquire a stretch of key and we shall see that given about 150 consecutive
key values the solution is relatively easy. Once again then we will see that
‘brute force’ estimates can be very misleading when it is a question of
assessing the difficulty of solving any cipher system. As we saw in Chapter