Attack on textbook RSA with small messages
The attacker sees $E(2^{100}\cdot m) = (2^{100} \cdot m)^e \,(N) = 2^{100e} \cdot m^e \,(N)$. The attacker can compute $A = 2^{100e} \,(N)$ from public parameters, so he can also compute $A^{-1} \,(N)$ (if not he would already have a factor of $N$). So he then knows $m^e \,(N) = E(2^{100}\cdot m) \cdot A^{-1} \,(N)$ and you can apply the previous attack. This abuses the homeomorphic property of RSA.