Using homomorphic encryption, I would like to be able to take an encrypted integer and either add 1 or -1 for a new encrypted value. I do not want the encrypted value to be recoverable - just the following:
- $\varepsilon(x) = (\varepsilon(x) \oplus \varepsilon(-1)) \oplus \varepsilon(1) = (\varepsilon(x) \oplus \varepsilon(1)) \oplus \varepsilon(-1),$
- $\varepsilon(x+1) = \varepsilon(x) \oplus \varepsilon(1) \ \mathrm{and}$
- $\varepsilon(x-1) = \varepsilon(x) \oplus \varepsilon(-1)$
- $\forall x \in \mathbb{Z}$
$x$ can never be recovered because the secret key will not be known (an original $\varepsilon(x')$ is chosen as $0$ and it will only be manipulated or used in encrypted form). It is ok (and perhaps even desirable) if $\varepsilon(x + n)$, computed (only) recursively as a series of single increments or decrements, is worse than $O(n)$ but no worse than about $O(n \log n)$ and there should be no limit on $n$ - that is, the further I go in a direction, it gets a little harder to keep going, but not impractically so.
It seems to me that this is perhaps within practicality for Gentry's scheme if the hints for $\mathrm{sK}$ can be reused forever for $\mathrm{recrypt}()$, but I might be misunderstanding this (or all of it). Then, Gentry might be overkill, since I don't seem to need multiplication ($\varepsilon(-1)$ does not need to be calculated as $\varepsilon(1) \otimes \varepsilon(-1) $, right?). I think any PHE supporting unlimited additions is good enough. Gentry's numbers "stiffen" - but is that the case with all systems? If they stiffen at about $O(n \log n)$ that's what I'm hoping for worst case.
I'd like to implement this in F# but 1) I need a good choice of algorithm supported by .NET or with an implementation in F#, and 2) How can I disable padding in such a scheme which normally one would be crazy to want to do?
The purpose is to create a dimension whose origin is unknown, but which can be navigated relatively. I'm below novice at math and latex and formalisms so if my attempts above are totally asinine, I apologize. Do you recognize an equivalent scheme that doesn't require HE?
