How can one calculate the key-space for a restricted substitution cipher? A restricted substitution cipher is one where no letter is assigned to itself. For an alphabet with 26 letters how many keys are there for a restricted substitution cipher? What is the correct way to approach this question? I have been adviced to look into cycle notations, Markov chains and the structure of the symmetric group and its conjugacy classes.
2026-03-25 02:58:57.1774407537
Key-space for a restricted substitution cipher
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A key for a substitution cipher on an alphabet $A$, is just a permutation of $A$ so there are $a!$ many keys in total (setting $a = |A|$, the size of the alphabet). With the restriction that no letter is mapped to itself we get that a key must be a so-called derangement.
As the linked page explains, the number of those on an alphabet of size $a$ is
$$a!\sum_{i=0}^a \frac{(-1)^i}{i!} \approx \lfloor \frac{a!}{e}\rfloor $$
so has the same order of size as $a!$.
For a $26$ letter alphabet the full key space size is $403291461126605635584000000$ and the restricted size is $148362637348470135821287825$ according to Wolfram alpha.