Given
- two large primes $p$ and $q$,
- two arbitrary odd integers $a$ and $b$,
- $x \in Z_{n^2}^*$ where $n$ = $pq$,
find whether there exists numbers $c$ and $d$ that are distinct elements in $Z_{n^2} \setminus \{0,1\}$ such that $x = c^a.d^b \mod n^2$
I assume the problem is called higher residuosity problem if we ignore $b$ and $d$ and is easy to solve given the factorization?