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I am new to cryptography esp, homomorphic encryption. I would like to know if the following is possible using something similiar to additive homomorphic secret sharing. Please apologize me if its too vague. I will try my best to explain my ask.

Alice has a secret that she wants to homomorphically encrypt. However Alice does not have enough resources to perform FHE. Hence Alice reaches out to a quorum of powerful devices, that say theoretically, can encrypt the data homomorphically for Alice. However since Alice cannot share the secret directly with the quorum, she decides to use a secret sharing scheme. But if she splits the secret directly and shares it, then a few members of the quorum can combine their shares and reconstruct the secret. Hence Alice wants to perform some operation on the secret and split it into shares for each of the member of the quorum such that even if the quorum members combine their share they wont be able to reconstruct the secret. But if they homomorphically encrypt their share ( or perform some operation on their share and then encrypt their share homomorphically ) and then combine their homomorphically encrypted shares, they will be able to produce the homomorphic encrypted cipher of the secret that Alice had.

Thanks in advance !

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    $\begingroup$ What problem are you really trying to solve? This feels like an attempt to solve a problem. It may be better to tell us the actual problem. $\endgroup$ – mikeazo Apr 24 '18 at 11:58
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    $\begingroup$ what about using "Hybrid fully homomorphic Framework", Alice gives an FHE encryption of a symmetric key (1 time setup), and Alice then gives a symmetric encryption of the secret she wants to send. Then the quorum hmomorphically decrypts the ciphertext and gets an encryption of the secret Alice had $\endgroup$ – Florian Bourse Apr 24 '18 at 13:25
  • $\begingroup$ @mikeazo :I am trying to completely outsource homomorphic encyption for a resource constrained device to a powerful device or quorum of powerful devices . However the communication between the resource constrained device and quorum cannot be in plaintext. Hence if the resource constrained device performs symmetric key encryption say AES and sends the encrypted data , from homomorphic evaluation of AES, I assume we can get homomorphic encryption on the original data. But this needs homomorphic encryption of AES key. So I am trying to figure out a way to homomorphically encrypt this AES key. $\endgroup$ – shrram Apr 24 '18 at 16:31
  • $\begingroup$ @FlorianBourse : Thanks for the pointer.. I think I looking for a way to get around Alice from doing the one time setup homomorphic encryption. I have updated the actual problem above. If you have any thoughts on this actual requirement, do let me know ! $\endgroup$ – shrram Apr 24 '18 at 16:34
  • $\begingroup$ How certain are you that the homomorphic encryption is so expensive that the resource constrained device cannot even run one encryption operation? $\endgroup$ – mikeazo Apr 24 '18 at 16:35

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