What is additive homomorphic encryption and multiplicative homomorphic encryption?

Question

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 ?

Answer 1

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.