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I am looking for a hash methodology that allows previous hashes to provide "hints" to find same or nearby hashes:

  1. if I expect to find a message $M = P + U$ consisting of private $P$ and public $U$ that hashes to some $H \lt N$ on average after $T = S/N$ tries, where $S$ is size of hash space (ie a cryptographically secure hash that can only be brute-forced)
  2. then there is a way to find another $P'$ in message $M' = P' + U$ that hashes to the same or "nearby" $H$" much easier, so that it takes $T' \ll T$ tries to find, by using the previous $U$ and $H$ (but not $P$ or $M$). But that, without $U$ and $H$ as hints, it takes $T$ tries same as before.
  3. The ratio between $T'$ and $T$ is configurable from parameters to the hash function (controlling how useful the "hint" is)

Does this make sense? Is this possible?

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    $\begingroup$ Might I ask what is the problem you're really trying to solve? It sounds like you're trying to come up with a puzzle or proof-of-work system - there is certainly a large amount of work already in those areas... $\endgroup$
    – poncho
    Oct 6, 2021 at 23:20
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    $\begingroup$ In $M = P + U$, what is operator $+$? Is it concatenation of bitsrings, bitwise exclusive OR, or some kind of addition reminiscent of that in $\mathbb Z$? What are the security requirements on $H$? $\endgroup$
    – fgrieu
    Oct 7, 2021 at 13:11

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