# Is a simple stream cipher “partially homomorphic” if no integrity check is applied?

My understanding is that, simply put, a stream cipher is just a CSPRNG such that $R(i,k)$ will produce a deterministic but statistically random sequence, where $i$ is an IV, and $k$ is the session key. The resultant key-stream $K$ is then combined with plaintext (i.e. $C = K \oplus M$).

Now, if we take two messages $m_1$ and $m_2$, and assume $\mathcal{E}(x)$ and $\mathcal{D}(x)$ are functions that encrypt and decrypt a message as described above, surely the following is a partially homomorphic cryptosystem?

$$\mathcal{D}(\mathcal{E}(m_1) \oplus m_2) = m_1 \oplus m_2$$

This seems a really simple and fundamental way to perform unauthenticated homomorphic cryptography, yet the Wikipedia article doesn't mention it at all.

Am I incorrect in saying that this process demonstrates homomorphism?

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To answer your question: no this is not homomorphic encryption because one of the plaintexts is used unencrypted. There may be times when it is a useful property, but the only uses I know of it are to demonstrate the malleability of xor ciphers.

To be a homomorphic encryption function, it should be possible to calculate the encryption of some function of the plaintexts from their respective ciphertexts without using the key. That is, given $c_1=e_k(m_1)$, $c_2=e_k(m_2)$ one should be able to calculate $c_3=e_k(f(m_1,m_2))$ by evaluating some $g(c_1,c_2)$.

More generally, a homomorphism is a mathematical function that preserves some operation. For example, if an encryption scheme $e$ would be xor-homomorphic if $$e(m_1)\oplus e(m_2) = e(m_1\oplus m_2)$$

More complex schemes might commute with other operations. For example the Pallier scheme uses the fact that $$a^x \cdot a^y = a^{x+y}$$

To construct a scheme that uses multiplication of ciphertexts to calculate the sum of plaintexts.

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This makes sense. I think malleable was the term I was originally trying to think of to describe this property. –  Polynomial Dec 4 '13 at 14:52
was going to edit this in, but since its more a side-point: An efficient Fully Homomorphic Encryption scheme (FHE) is in many ways the holy grail of cryptography, since it would allow encrypted evaluation of any circuit - which basically means you can secretly evaluate any function. –  figlesquidge Dec 4 '13 at 17:40