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1

I know of a proposal from Matsuo in 2007, called "Hybrid Proxy Re-Encryption", which allowed to transform PKE ciphertexts to IBE ciphertexts. This proposal is designed for a specific PKE and IBE scheme, not for generic ones. Note also that this is not what we usually understand by the term "proxy re-encryption", which is devoted to transforming ciphertexts ...

1

I think there is no trivial or simple way of performing divisions between two ciphertexts. However, for several applications, as calculating mean and standard deviation (as you wrote in your question), it is sufficient to do divisions between a ciphertext and a plaintext, because the denominator is known (in those cases, it is $N$, the number of elements ...

1

In case some assistance from plaintext holder is available: (1) choose a random, multiply it with the plaintext and also homomoprhic-encrypt the random; (2) calculate inverse of the product in clear (no encryption); (3) multiply encrypted random by the inverse. This is not exactly division, still it outputs the inverse.

4

Using generic homomorphic encryption the answer for all three is essentially yes. Although 3. in general is probably mainly of theoretical interest as it would be impractically slow. For simply compressed data this is simple since compression should not do anything to hide the data. Just decompress, do the operation and possibly compress the result if ...

0

There is a way to use ECEG that gives us an additive homomorphic property, however the addition in this case is Elliptic Curve point addition, not addition over the integers. One way to use ECEG to encrypt a point $M$ is to have the encrypted value be the pair of points $(kP, M + kY)$ That way, if you have the encryption of the points $M, M'$ as $(kP, M + ... 0 Have a look at order revealing encryption (ORE) schemes: https://eprint.iacr.org/2014/834.pdf 1 You'll have to write the function you are trying to calculate as a polynomial in the two inputs$x$and$y$. If you are working over the field with$q$elements as plaintexts, you have to calculate for equality the polynomial$(x-y)^{q-1}\$. Greater-than-or-equal (however you define that for finite fields) will be even more complicated.

2

I think the keyword equality can be done with FHE, but, as you pointed out, the server will not know the result of the test since it will be encrypted. Considering this, the answer to your question "Does this mean server does not know whether the string matches the document?" should be yes, it does. The server does not know whether a match occurs or not. ...

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