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I am trying to convert a homomorphically encrypted integer to a homomorphically encrypted binary number (in vector form like {Ciphertext(1),Ciphertext(1),Ciphertext(0),Ciphertext(1)}), which I cannot really get the remainder (and any other "intermediate results") without decryption.

Is there really a way to do this?

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  • $\begingroup$ That may depend on the FHE scheme that you are using. Generally, that is too way hard to do. Better way is to use binary in the beginning! $\endgroup$
    – kelalaka
    Commented Aug 10, 2020 at 14:38
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    $\begingroup$ FHE is by definition Turing Complete HE, so it's certainly possible there. But not all HE is FHE. $\endgroup$ Commented Aug 10, 2020 at 16:24
  • $\begingroup$ Which homomorphic scheme are using? How are you encrypting the messages? Simply one message per ciphertext or using some batching technique to encrypt several messages into one ciphertext? $\endgroup$ Commented Aug 11, 2020 at 6:28

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The question could be change to "How to evaluate mod $N$ under FHE". For CKKS scheme, there exist approximations for that: https://eprint.iacr.org/2018/153.pdf

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