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If you consider arbitrary permutations, you have $\frac{n(n+1)}{2}$ possibilities. That means, $O(n^2)$ is the correct complexity in big-O notation, but I don't understand why you need that at all, if you can provide the result as exact formula. Caesar cipher contains just a subset of $n$ possibilities, and therefore obviously $O(n)$. Anyway, this doesn't ...


How on earth did you arrive at that formula? You can break a Caesar cipher by calculating the result of applying all of the $n-1$ (i.e., 25) possible shifts to the ciphertext and picking the one that makes sense. The computational complexity is just $\mathcal{O}(n)$. If you want to automate the process based on frequency analysis, the correlation step ...

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