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

Since some cages are obviously very ‘bad’ (that is, weak from the cryptographer’s
point of view) this raises the questions:
(1) How many possible cages are there?
(2) How many of these are ‘good’ cages?
The Hagelin cipher machine 141
We can find the number of distinct possible cages exactly since the
number is given by
number of possible cages�the number of representations of 27 as
the sum of six non-negative integers
and although there is no simple formula for calculating this number we
can use a nice mathematical identity and a computer to discover that the
number is 811. If we insist that each wheel must have at least 1 lug opposite
to it, because a wheel with no lugs opposite it might as well not be
there, the number reduces to 331. The same technique reveals that if we
only use 26 or 25 lugs in the cage the number of possibilities, allowing 0
lugs opposite one or more wheels, reduces to 709 and 612 respectively, or
to 282 and 235 when we disallow 0 lugs. We must remember however
that since the six numbers of a cage may be permuted and so generate different
key streams, although these streams will have similar statistical
features, the number of possible cages that the cryptanalyst must consider
is substantially larger. Thus, in the case of 27 lugs there are 201 376
possibilities, not 811; for further details see M18.
How many of these 331 distinct possible cages are ‘good’? Clearly it
depends upon what we mean by ‘good’. If we take it to mean
a ‘good’ cage is one that generates all possible keys in the range 0 to
25
then we can use a computer to find how many of the 331 do this. The
answer is 113. Permuting the cage makes no difference to its quality: a
‘good’ cage remains ‘good’ and a ‘bad’ cage remains ‘bad’. If we allow 0
lugs the 881 possible cages yield only 120 ‘good’ cages; in other words,
only 7 cages with a wheel with 0 lugs produce all 26 possible key values
(mod 26). Of course even a ‘good’ cage may be ruined by highly asymmetric
pin settings, such as 90% active, 10% inactive.
Problem 10.2
Determine which of the following 6 cages generate all key values, 0 to 25
(mod 26). For those cages which fail to do this find which key values are
missing.