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

ways, since choosing to pair, say, A and W is the same as choosing to pair W
and A. Similarly the second pair can be chosen in
24�23
2
ways, and so on. There would therefore seem to be
26�25�24�23�•••�4�3�2�1
2�2�•••�2�2
ways of forming reciprocal alphabets but this is not so, for we can rearrange
the 13 pairs in any order without changing the substitution
alphabet. For example, if we choose to pair A and W and then to pair B and
K we would get exactly the same result as if we first paired B and K and
then paired A and W. We must therefore reduce the number above by the
factor
13!�13�12�11�•••�2�1
which is greater than 6227 000 000 and since the 13 factors of 2 in the
denominator above provide a further factor of 8192 we see that we have
reduced the number of substitution alphabets by a factor of more than
50 000 000 000 000. Overall this means that the number of possible substitution
alphabets is reduced from more than 10 to the 26th power to
fewer than 10 to the 13th power.
It may seem strange, but it is better not to pair all of the 26 letters; the
number of possibilities is increased if only 22 are paired and the other 4
letters left unchanged. This is because the number of possibilities if we
pair 2k letters and leave (26�2k) unchanged is
(26!)
(k!)((2n � 2k)!)2 k
Mathematical aspects 191
and this reaches a maximum at k�11. Whilst this fact is not important for
simple substitution ciphers it is of significance in the context of the
number of pairings on the Enigma plugboards, as we shall see in Chapter
9.
M3 The birthdays paradox
The probability that two people chosen at random have the same birthday
is 1/365. We ignore leap years, which have no significant effect on the
result.