I need to find all the involutory keys for the substitution cipher over $\mathbb{Z}_7$. I wasn't sure what can be the key for the substitution cipher. For example, for the affine cipher, $e(x) = kx + a$, $(k,a)$ is the key. However for the substitution cipher, I can't find the corresponding function.
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1I'd say the key is a permutation. So find all permutations of 7 elements that are involutions. – Henno Brandsma Jan 23 '16 at 15:29