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I wonder if there is a known hash function that for any $x$, $y$ which are numbers in a certain range with certain accuracy (I guess one can think of them as integers?), you can define two functions corresponding to addition and multiplication such that:

$h(x) \# h(y) = h(x+y)$

and

$h(x) \cdot\cdot h(y) = h(x \cdot y)$.

Edit: I now see it is slightly duplicated with Is it possible to subtract/multiply numbers using homomorphic encryption?

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    $\begingroup$ What security properties are you looking for this 'hash function'? Preimage, second preimage, collision resistance? Something that actually has a fixed output length independent of the input size? $\endgroup$ – poncho Oct 1 '19 at 20:58
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    $\begingroup$ It is not duplicate, they are not hash function. $\endgroup$ – kelalaka Oct 1 '19 at 21:08
  • $\begingroup$ I remember a similar question but couldn't find it. $\endgroup$ – kelalaka Oct 1 '19 at 21:24
  • $\begingroup$ I agree with poncho here; I don't see how you can have such a hash function without dropping at least some of the common properties of a generic cryptographically secure hash function. $\endgroup$ – Maarten Bodewes Oct 1 '19 at 21:38
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    $\begingroup$ Possible duplicate of Are there any practical implementation of a homomorphic hashing or signature scheme? $\endgroup$ – Squeamish Ossifrage Oct 1 '19 at 22:08

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