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Is it possible to choose such a nontrivial set of queries so that the amortized running time for a hash table with a public key and a function like $(a_{N − 1}k^{N − 1} +... + A_{1}k + a_0) (mod \: p)$ for $N = 2$ is $\Omega(\sqrt{n})$, where $n$ is the total number of keys in the table (as a last resort, at least $\Omega(\log n)$)?

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    $\begingroup$ Where do the $a_i$'s lie, what is the domain of the hash function? Where does the public key come in? Is the hash chosen at random? Those details are crucial to being able to give a meaningful answer. $\endgroup$
    – Ariel
    Dec 1 '20 at 14:53

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