I have two different ciphertexts $c_1$ and $c_2$, and two different public keys that share the same modulus such that $e_1 \neq e_2$ but $n_1 = n_2$.

  • What attack can I use to recover the private keys?
  • Common modulus attack works only if $c_1 = c_2$, doesn't it?
  • 2
    $\begingroup$ This might be a homework question. $\endgroup$
    – kelalaka
    Commented Oct 22, 2019 at 19:36
  • 2
    $\begingroup$ Obviously the question is about textbook RSA. Also, something must be known about the plaintext(s); such that: they are the same. Otherwise, $c_1$ and $c_2$ give no information and the only way is factoring $n_1$. $\endgroup$
    – fgrieu
    Commented Oct 22, 2019 at 19:41
  • 4
    $\begingroup$ Actually, even with textbook RSA, there isn't a way to recover the private key, even if the messages were the same. You could recover the common message (assuming $e_1, e_2$ are relatively prime), and that's a standard textbook attack... $\endgroup$
    – poncho
    Commented Oct 22, 2019 at 21:55