EDIT: Rephrased.
I have, stored somewhere, the values $a$ , $Q$, $N_1$ (plus its factor) and $a^{2Q} \mod N_1$.
I also know $b$, $R$ and $N_2$ (but not its factors).
I want to know whether there is a calculation someone else can do (without knowing $b$, for example mixing it with $a$) such that the result will let me find $b^{2R} \mod N_2$ more easily than just exponentiating the usual way.
The other could have access to (I'll give to him in plain if necessary) the values $Q$, $N_1$ and $a^{2Q} \mod N_1$, but I must kept secret the value $a$ and $b$. Anyway, I can disclose any result made from an operation on $a$ and $b$ which is infeasible to invert.
Thank you