I know modulus $N$, public exponent $e$ and have some crypted messages $c_i$, $c_i = m_i^e mod N$. I know something about each $m_i$: they have a sequence of bytes BAADF00D at offset $t$.
Is there any kind of attack which helps me to recover any $m_i$ or even $p$ and $q$ used to make $N$?
I thought about an Oracle attack, but the papers I read discussed Oracles with MSB or LSB bits. What with bits in the middle of a message?