# Scheme where earlier hashes provide "hints" to find same or nearby hashes

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?

• 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... Oct 6, 2021 at 23:20
• 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 $H$?
– fgrieu
Oct 7, 2021 at 13:11