How multiplicative property of RSA can be exploited It is known that RSA has the multiplicative property

$E(m_1) \cdot E(m_2) = E(m_1 \cdot m_2)$

because

$m_1^e \cdot m_2^e = (m_1 \cdot m_2)^e  \pmod n$

But how can this property be used to break the RSA system?

What are the attacks that use this property?
Using this property, you can't fully break RSA (meaning you can't get the private key).

However there are a few attacks.


If you're using RSA for public-key encryption, there's an adaptive chosen ciphertext attack. Suppose you want to decrypt $c$ and A is decrypting everything but $c$ for you. Now you can chose your $c_1=x^e*c$ with $x\in Z_n^*$. Now you you send $c_1$ to A, he decrypts this. Now you'll have $m_1=c_1^d=c^d*(x^e)^d=m*x$ $(mod$ $n)$, now you simply compute $m=m_1*x^{-1}$ $(mod$ $n)$ and decrypted an arbitrary ciphertext
If you're using RSA for digital signature with message recovery you can basically mount the same attack as above, by basically using blind signatures so you can get arbitrary data signed. (If not using proper methods to protect)