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Explain in brief about additive homomorphic encryption and multiplicative homomorphic encryption. I have read about them but I could not properly understand. So would you please elaborate additive homomorphic encryption and multiplicative homomorphic encryption ?

user11236
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    You should respond to the comments on your last question before asking a new one. Also, this is not the type of question you should ask here. Instead of 'explain this', ask specific questions with specific answers. If there was something you did not understand when you read about these things, ask a question about the thing you did not understand. – Michael Albanese Jan 08 '14 at 02:38
  • ok Mr. Michael Albanese . Actually I didn't know that how I ask question to you. So just I abruptly asked the question in which I had doubt. But next time I'll keep in mind. – user11236 Jan 08 '14 at 02:58

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Multiplicative

If the RSA public key is modulus $m$ and exponent $e$, then the encryption of a message $p$ is given by $$\mathcal{E}(p) = p^e \;\bmod\; m.$$

The homomorphic property is then $$\mathcal{E}(p_1) \cdot \mathcal{E}(p_2) = p_1^e p_2^e \;\bmod\; m = (p_1p_2)^e \;\bmod\; m = \mathcal{E}(p_1 \cdot p_2).$$

This is saying that if we take two plaintext messages $p_1$ and $p_2$ and multiply them together and then encrypt that using RSA, we get some ciphertext. The multiplicative property is saying that you can also encrypt each plaintext separately and then multiply the two ciphertexts together and you get exactly the same thing. This is a remarkable property. Try it with a baby example of RSA and see it for yourself (use very small numbers).

Unfortunately, RSA is only half-fully-homomorphic-encryption (FHE) since the additive property does not follow.

There are other such examples of half-FHE cryptographic systems and a lot of research was being done to find a full-FHE.

From this multiplicative example, can you now figure out what additive is?

It took 30-years after the discovery regarding RSA, but Gentry found an example of an FHE (crypto.stanford.edu/craig/craig-thesis.pdf‎), however, it is not practical from a performance point of view.

Many people are trying to figure out how to do a practical FHE and various schemes have been proposed because this is very useful for cloud computing and security. Do you understand why?

In addition to this, you might want to investigate obfuscation, white box cryptography and functional encryption. The last is also partially by Gentry and shows promise as these all protect intellectual property from reverse engineering and all are rooted in abstract algebra, algebraic number theory, number theory and related areas.

Amzoti
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  • Thanks, What is the advantage and disadvantage of multiplicative homomorphic encryption ? – user11236 Jan 08 '14 at 03:24
  • You can store encrypted data in the cloud and perform transforms on it without decrypting it. Please recall to upvote and/or accept answers that are helpful. Regards – Amzoti Jan 08 '14 at 03:29
  • What is the disadvantages of multiplicative homomorphic encryption ? – user11236 Jan 08 '14 at 03:38
  • What is more secure between additive homomorphic encryption and multiplicative homomorphic encryption ? Why ? – user11236 Jan 08 '14 at 03:39
  • But if I want to use that in my research topic of "Provable Data Possession at Untrusted Stores" , so what will be the better option ? additive homomorphic encryption or multiplicative homomorphic encryption ? Why ? – user11236 Jan 08 '14 at 03:48
  • Well, since multiplicative choices are more readily available, I think the answer is obvious, else you would have a massive performance problem. See, for example: http://en.wikipedia.org/wiki/Homomorphic_encryption. – Amzoti Jan 08 '14 at 03:49
  • By the way, because something is provable secure does not mean it is practical because the real world adds all sort of real-world challenges! For example, look at many reports of flawed crypto and security protocols (many examples). Storing, transporting, unencrypting, memory, reverse engineering, side-channel attacks .... and all the rest are ever present. Regards – Amzoti Jan 08 '14 at 03:53
  • @Amzoti: I am so glad of being your friend. Amy and you. :-) – Mikasa Jan 09 '14 at 03:18
  • ohh ya happy new year – user11236 Jan 09 '14 at 16:29
  • actually I am doing research on Network Security so I just wanted to know whether additive homomorphic encryption or multiplicative homomorphic encryption is better . – user11236 Jan 09 '14 at 16:30
  • anybody know about dynamic provable data possession ? – user11236 Jan 09 '14 at 16:35
  • What is the difference between homomorphic encryption and homomorphic signature ? – user11236 Jan 12 '14 at 06:37