Using LUC RSA, I have a list of 8 public keys $(R_1,n_1)$ through $(R_8,n_8)$, and a list of $8$ messages $M_1\dots M_8$. The messages have all been double encrypted. However, one of the public key exponents, $n_5$, is not given. The problem is to decrypt all of the messages, and so I must also determine the value of the missing $n$. The hint is that the exponent was constructed "poorly".
Having factored all (including $R_5$) but two of the $R$s, I have decrypted $5$ of the messages.
My question is, what exactly is a poorly constructed public exponent look like, and how can I use what I have been given to discover the missing $n$?
Given the length of one particular encrypted message ($M<R$), it is likely that it was encrypted the last time with $R_5$.
So, I suppose what I'm asking is: Can you solve for n given $E=M^n \mod R$ where only R (public modulus) and E (ciphertext) are known? I suppose that's basically the DLP...