# How many known plaintexts are required to break a system with this property?

Suppose we have an $$n$$ bit block cipher $$E$$ with a key $$k$$ that has the following property $$E(k; m_1⊕m_2) = E(k;m_1) ⊕E(k; m_2).$$ How many minimum number of chosen plaintexts are required to decipher (with probability 1) a given ciphertext (not from the list of ciphetexts corresponding to the chosen plaintexts) without knowing $$k$$ and what would be those plaintexts be.

I figured replacing $$m_2$$ with a n-bit string of all 1's can give us $$E(m')=E(m)'$$ where $$m'$$ is one's complement of the bit string.

I cannot think of anything further than this. Not knowing the encryption algorithm seems to be a hurdle, although my teacher said it should not be.

• And, is this homework? We call this homomorphic encryption. An example is Golderwasser-Micali – kelalaka Sep 15 at 18:43
• @kelalaka: "although my teacher said it should not be" - I assume that it is either homework or a classroom exercise... – poncho Sep 15 at 18:44
• Hint: Find two messages that their encryptions' x-or is the target ciphertext! – kelalaka Sep 15 at 18:49