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Is there any public/private homomorphic encryption scheme that directly works on floating numbers or vectors of floating numbers?

In our application, we want to find out if $$v_1 \approx v_2, \qquad\text{or more precisely}\qquad \lVert v_1-v_2\mathbin\rVert < \epsilon,$$ with $v_1$ and $v_2$ being vectors with a fixed number of floating numbers.

An additional requirement for privacy protection is that we don't want the application to have access to the raw $v_1$.

So we suggest encrypting $v_1$, $$w_1 = \mathit{Enc}(v_1, k_{\mathit{pub}})$$ and sending the encrypted version $w_1$ and the public key.

In the application, we can then check if $$\mathit{Enc}(v_2, k_{\mathit{pub}})\approx w_1, \qquad\text{or more precisely}\qquad \lVert w_1-\mathit{Enc}(v_2, k_{\mathit{pub}})\rVert < \delta.$$

Is there an encryption scheme that supports this application?

Some references I found:

Squeamish Ossifrage
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eavsteen
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  • There is mix on more precise part. Also, who is going to access the result? – kelalaka Oct 08 '19 at 12:32
  • Bob is encrypting its $v_1$ and sending $w_1 = Enc(v_1, k_{pub})$ and $k_{pub}$ to Alice. Alice only wants to check if its $v_2$ is the same to $v_1$ or not. We want to make sure if Alice is compromised that the attacker doesn't have access to the raw $v_1$. – eavsteen Oct 09 '19 at 07:59
  • So, Bob uses his public key then, Alice cannot learn the output since it encrypted. If Bob uses Alice's public key than Alice can learn the $v_1$ – kelalaka Oct 09 '19 at 08:02
  • Yes but in this application we only want Alice to check if her key is the same as Bob's. Bob is only sending its encrypted key to Alice for Alice to check. We want to prevent Alice to learn about $v_1$, so that if she is compromised then the attacker does not have any access to $v_1$. – eavsteen Oct 09 '19 at 08:14
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    You need something like the socialist millionaire problem. This might be helpfull – kelalaka Oct 09 '19 at 08:21
  • Does it work for $v_1 \approx v_2$? It seems that it only works for $v_1= v_2$. So I think it is the socialist millionaire problem with a twist. Two millionaires want to check if they have approximately the same wealth. – eavsteen Oct 09 '19 at 08:49
  • Bob (application) can for sure encrypt $v_2$ and compute the distance $|| w_1 - Enc(v_2, pk)||$, Bob can even implement a circuit that test if this distance is smaller than $\delta$, but he can't see the result of that test, because if he could, that would leak information about $v_1$, and homomorphic computations do not leak information about the plaintexts. – Hilder Vitor Lima Pereira Oct 16 '19 at 06:42

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