The goal of this question is to allow a server/proxy to forward an encrypted message without being able to read it with this procedure being transparent to the original sender and receiver.
Assume we have Admin RSA Keys $(e,d,n)$ and Bob RSA Keys $(e_b,d_b,n_b)$ where Alice wants to send a message $m$ to Bob with only knowing the Admin's public key $(e,n)$. She sends $c_0 = m^e \mod n$ to a proxy where $c_0$ is transformed to $c_1$ which is forwarded to Bob who decrypts it as $m = c_1^{d_b} \mod n_b$.
The magic is in the transformation from $c_0$ to $c_1$ without giving the proxy the Admin's private key $d$.
My thinking here is that if (!) $n = n_b$ we could compute a re-encryption key $$ r = d \cdot e_b $$ in advance. If we also could protect $r$ in such a way that the proxy is unable to retrieve $d$ from $r/e_b$, I believe we could simply compute $$ c_1 = c_0^{r} \mod n_b $$
These are a lot of assumptions that will not likely hold in practice. Especially, the biggest difference will be that $n \neq n_b$.
So the question is, can these obstacles be overcome with modulo arithmetic or is there another way to create a re-encryption (pseudo-)key in RSA that prevents retrieval of $m$ while transforming $c_0$ to $c_1$.
