Multi-party RSA with Small e

Assuming $e$ is constant between the messages and the message $m$ is sent to at least $e$ people, we can use the Chinese Remainder Theorem to retrieve $m$.

In single-party RSA, we calculate $c = m^e \mod N$. Let's pretend this is extrapolated to 3 people:

$$m^e = c_1 \mod N_1 \\ m^e = c_2 \mod N_2 \\ m^e = c_3 \mod N_3 \\$$

​The Chinese Remainder Theorem allows us to solve this congruence $\mod N_1N_2N_3$. Since $m < \min{N_1, N_2, N_3}$, we know that $m^e < N_1N_2N_3$. Once we use the Chinese Remainder Theorem to compute $m^e \mod N_1N_2N_3$, we just take the $e$th root to retrieve $m$.​