Encrypting a small value $m\in\{1,\dots,100\}$ using textbook RSA When encrypting a small value $m\in\{1,\dots,100\}$ using textbook RSA, and since that is deterministic, an attacker could try every possible number to determine $m$.

To prevent this kind of attack, one could select a big random number $x$, and send $x^e\bmod n$ and $(m+x)^e\bmod n$.

What would be a problem with this encryption technique?
If $e$ is "small" (as all typical values of $e$ are!), this is vulnerable to the Franklin-Reiter related message attack.

In a nutshell, given $c_1=x^e\bmod n$ and $c_2=(x+m)^e\bmod n$, for each possible value of $m$, compute
$$f:=\gcd(y^e-c_1,(y+m)^e-c_2)$$
in the polynomial ring $(\mathbb Z/n)[y]$. If you guessed the right $m$, then (possibly except for very rare cases)
$$f=y-x\in(\mathbb Z/n)[y]$$
and you have recovered $x$ and verified that $m$ was correct. Otherwise, $f$ will most likely be $1$. Note that this $\gcd$ is technically not well-defined in all cases, but if you actually encounter a problem with that, you have effectively factored $n$.

Of course, this assumes that $e$ is small enough for this $\gcd$ computation to be feasible, and that the set of possible messages $m$ is small enough to be brute-forceable.