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As I understand from this article about homomorphic encryption, it mainly aims to enhance the security of cloud computing. We should be able to encrypt data and send it to the cloud. After it is send, we should be able to make computation on the encrypted data and the result of this computation is an encrypted data too. If we decrypt the result of our computation, then we should get back the plain text version of the result.

The question is, why we assume it enhances the cloud computing? We could just encrypt data, send it to cloud. When we need to make computation, query these data to our computers and decrypt it, make computation, then send it back to the cloud if needed.

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Homomorphic encryption doesn't "mainly aim to enhance the security of cloud computing." Homomorphic encryption happens to enhance cloud computing security, but that wasn't its aim when it was developed. –  mikeazo Mar 3 '13 at 13:31
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Your solution is not "cloud computing", just "cloud storage". –  Paŭlo Ebermann Mar 3 '13 at 21:55
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Homomorphic encryption in and of itself solves the problem of computation on encrypted data. For example, if I give you only the encryptions of $a$ and $b$ ($E(a),E(b)$) can you compute $E(ab)$ and $E(a+b)$. That is the problem that fully homomorphic encryption (FHE) solves. There are also partially homomorphic encryption methods that allow you to only compute one operation (e.g., Paillier and Elgamal).

Cloud computing is "the use of computing resources (hardware and software) that are delivered as a service over a network (typically the Internet)." So, you push your data to the cloud then tell the cloud what computation to perform. The cloud computes the result and sends the answer back to you. The benefit of FHE when applied to the cloud is that all of this can take place without the cloud learning your data. You upload encrypted data, the cloud performs the comptuations on that data, returns the result. That said, it appears that FHE alone is not sufficient for privacy-preserving cloud computing.

Of related interest, I'd suggest you look at In what ways does Full or Partial Homomorphic Encryption benefit the cloud?

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It beats the very purpose of moving to cloud , its not always possible to download entire data pushed to cloud , decrypt it and perform operations on it and again encrypt and send it back.

One of the key advantages of cloud computing is massive scale and elasticity . imagine peta bytes of data being moved to cloud . it beats this very purpose to download all to do some computations on it ,

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Also the cloud is not trusted. The cloud might acts maliciously so this is the reason you encrypt with a symmetric key cipher your data before uploading them. Still fully homomoprhic encryption as far as we know costs a lot and there are practical solutions only for specific operations on data

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