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This precise issue recently arose in light of suspicious patterns in the S-box of a Russian cipher Kuznyechik. See: Xavier Bonnetain and Léo Perrin and Shizhu Tian: Anomalies and Vector Space Search: Tools for S-Box Analysis, Asiacrypt 2019 One way the authors chose to quantify how unlikely such a permutation could have occurred by chance is to find the ...


There are at most $n \cdot (n - 1)$ permutations of $\mathbb Z/n\mathbb Z$ of the form $x \mapsto ax + b$: if $n$ is prime, there are $n - 1$ choices for $a$ and $n$ choices for $b$ under which this is a permutation. There are $n!$ permutations of $\mathbb Z/n\mathbb Z$ altogether. So the probability that a uniform random permutation has this form is ...

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