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

Appendix
Mathematical aspects
Chapter 2
M1 Identical letters in substitution alphabets
This, the problem of the number of derangements of the elements of a set,
follows as a special case of what is known both as the classical sieve formula
and as the inclusion and exclusion principle, proofs of which can be found in
books on combinatorics such as [2.6]. The sieve formula tells us that the
fraction of the number of permutations of the numbers (1, 2, ..., n) which
are such that no number is in its ‘correct’ position, i.e. of derangements of
n numbers, is
1 1 1 1 1 1
� � � � � �••• for (n�1) terms
1 1! 2! 3! 4! 5!
and this fraction rapidly converges to the value 0.3678... which is the
reciprocal of the number e, for those familiar with natural logarithms. For
n�0 to 6 the fractions have the values to three decimal places 1, 0, 0.5,
0.333, 0.375, 0.367, and 0.368. Thus the fraction is virtually the same in
practical terms for values of n greater than 5. This means that a permutation
alphabet on 26 letters has approximately a 37% chance of having no
letter in its ‘correct’ place and hence a 63% chance of having at least one
letter in its original position.
M2 Reciprocal alphabets weaken security
We can choose the first pair of letters in
[190]
26�25
2