Hash function over numbers that preserve addition and multiplication

Asked Oct 01 '19 at 20:49

Active Oct 01 '19 at 20:49

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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?

asked Oct 01 '19 at 20:49

Uri

3

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? – poncho Oct 01 '19 at 20:58
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It is not duplicate, they are not hash function. – kelalaka Oct 01 '19 at 21:08
I remember a similar question but couldn't find it. – kelalaka Oct 01 '19 at 21:24
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. – Maarten Bodewes Oct 01 '19 at 21:38
4

Possible duplicate of Are there any practical implementation of a homomorphic hashing or signature scheme? – Squeamish Ossifrage Oct 01 '19 at 22:08
@SqueamishOssifrage Would it be an issue that there are two operations required in this question? I presume that you could perform repeated addition for smaller numbers, but that would not be the same as fast multiplication... – Maarten Bodewes Oct 02 '19 at 00:00
I think finding preimages could be easier than the generic attack. However, the question is vague. One really needs to define the operation on the range of the hash. The addition has only one carry but multiplication doubles the output. To analyze one has to define the function properly. – kelalaka Oct 02 '19 at 08:05

Hash function over numbers that preserve addition and multiplication

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