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Hash function producing cycles with expected max length

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Marcin Król

Is there a known hash function $H_k: X\to X$ such that: $\forall{x\in{X}},\exists{n\in{\mathbb{N}}}, n<k \land H^n(x)=x$

=== EDIT ===

By hash function I mean that any other way of finding the preimage of $x \in X$ than iterating $H_k$ is computably unfeasible or at least significantly harder.

My motivation is using such a function as sequential POW.

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fgrieu avatar
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fgrieu
Exactly what definition of "hash function" are you using? If we ignore "hash", an obvious example is $H_k$ the identity function. Hint: prove that any such $H_k$ is a permutation of set $X$, thus perfectly collision-resistant, and that the (first and only) preimage of any $x\in X$ can be found with at most $k-1$ evaluations of $H_k$, which limits preimage resistance. But if $k$ and $|C|$ can grow exponentially with the security parameter for preimage-resistance, perhaps we can construct a candidate $H_k$. For a complete answer, convince us this is not homework, or you worked on it.
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