I want to update the secret key of RSA.
I understand there exists a common modular attack on RSA. Precisely, if adversary knows $(e_1, m^{e_1})$ and $(e_2, m^{e_2})$, where $e_1$ and $e_2$ are the RSA exponents under the same module $N$. Then it can compute $\lambda_1, \lambda_2$ such that $\lambda_1e_1 + \lambda_2e_2 = {\sf gcd}(e_1, e_2)$, where ${\sf gcd}(\cdot)$ is the greatest common divisor. Thus the plain message can be recovered by computing $(m^{e_1})^{\lambda_1/{\sf gcd}(e_1, e_2)}\cdot (m^{e_2})^{\lambda_2/{\sf gcd}(e_1, e_2)}=m$ (if I am wrong, please correct me).
In my implementation, I want to use the RSA like a private-key encryption. Instead of publishing the public RSA exponent $e$, I keep it secret. Then, the secret key is $(e, d)$ where $ed = 1 \mod \phi(N)$ (I do not consider to update the prime $p$ and $q$). Initially, I compute the RSA ciphertext $m^{e_1}$ and store the secret key $(e_1, d_1)$. At some time, I update the secret key to $(e_2, d_2)$ and give $d_1e_2$ for re-encrypting the ciphertext from $m^{e_1}$ to $m^{e_2}$.
My question is, 1) In my implementation of using RSA like a private-key encryption, the key-update is still vulnerable to common modular attack (or is it secure)? In other words, if the adversary knows $m^{e_1}, m^{e_2}$ and $d_1e_2$, is it able to guess $m$? 2) If 1) is secure, the key-update is resistant against leaking old secret keys? In other words, if the adversary knows $m^{e_2}, d_1e_2$ and $(e_1, d_1)$, is it able to guess $m$?
Waiting for your answers. Thanks a lot.