Attack on textbook RSA with small messages We learned in class that if Alice wants to send a small message $m$, such that $m &lt; N^{\frac{1}{e}}$, so we can break the system using $e$-th square of $m^e$.

But what if Alice knows this flaw and decides to sent $2^{100}\cdot m$ as her message. Is an attacker can still break the system (I assume that the attacker knows that Alice sent $2^{100}\cdot m$)?
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.