Endless Emails

We are given multiple messages, but from the hint, some of the messages are repeated. Hence, it is likely some (if not all) of the messages will be the same. That is, there is a message $m$, moduli $n_i$, and ciphertext $c_i$ such that: $$ \forall i, c_i \equiv m ^ 3 \mod n_i $$ Hence, we can employ Chinese Remainder Theorem in this case. This is also a simplified form of Hastad’s broadcast attack....