I am struck with the following problem:
Let Alice, Bob, Chris and Eve communicate over a public network. They encrypt all messages they send using RSA system. Bob and Chris have the RSA modulus $n_B$ and $n_C$ respectively with $n_B$ = $n_C$ But different public encryption exponents: $e_B \neq e_C$. Suppose $\gcd(e_B, e_C)=1$, and that Alice sends the same secret message to Bob and Chris.
Show how Eve can decipher the message.
Think about this: what does it mean that $\gcd(e_B, e_C)=1$. Formally that means there exist some $s_1, s_2$ such that $e_Bs_1 + e_Cs_2=1$.
Say you have two ciphertexts (the following math is all done modulo the shared modulus), $C_B=M^{e_B}$ and $C_C=M^{e_C}$. You can do the following:
$$\begin{align} C_B^{s_1}*C_C^{s_2}&=(M^{e_B})^{s_1}*(M^{e_C})^{s_2}\\ &=M^{e_Bs_1}*M^{e_Cs_2}\\ &=M^{e_Bs_1+e_Cs_2}\\ &=M^1\\ &=M \end{align}$$
