I learned about the RSA method, where if B wants to send a message $M$, say $0 \leq M <n = pq$ to A with public key $(n,e)$, then B sends $M'= M^e (mod \ n)$. Then A can decode this message using private key $d$ chosen such that $ed \equiv 1 \ mod \ ((p-1)(q-1) )$.
I can see that once someone figures out what the factorization of $n$ is, then it is easy to crack it. And I know that if $n$ is large then factoring is difficult.
If we wanted to decode $M'$, even if we don't know the factorization, we could just let the computer run to compute $x^e (mod \ n)$ for each $x$ until it matches $M'$. I thought maybe this could be done faster than finding the factorization of $n$, but I wasn't really sure. Is it as difficult (or time consuming) as factoring $n$? Could someone please explain to me this part?
Thank you very much!!
For typical key sizes, there are way too many possible plaintexts to encrypt to create a complete table in practical time periods. We are usually talking about numbers on the order of $10$ raised to three- or four-digit exponents. Even if encrypting a plaintext was a single machine operation, even $10^{100}$ messages takes too long to create a complete table. It would be far easier (though of course far from being actually easy!) to approach the problem by factorization.