Let us consider a product $P$ (whose factors we do not know). Given a base $b$, such that all the Modulus Residues are calculated using powers of $b$ w.r.t. $P$.
For some (unknown) power $x$ we know the corresponding residue ($R_x$), i.e.:
$b^x\bmod P = R_x$.
Query 1: Without knowing (or calculating) the factorization of $P$ and value $x$, for a given $k$, (using $R_x$) is it possible to calculate $b^{(x-k)}\bmod P = R_{(x-k)}$ efficiently?
Query 2: Can the above procedure (if known/exists) help in factorization of $P$?
I am uncertain about how to achieve 1 and its relation to factorization if any.