Let $n = P\cdot Q$ be the product of two safe primes $P = 2p+1$ and $Q=2q+1$. Let $g$ be a generator of $C_{p} \subset \mathbb{Z}_n^*$, the multiplicative subgroup of order $p$. In other words, $g^p = 1 \pmod n$. (But $p$ is still secret of course.)
Would it weaken a RSA modulus if $g$ was public? It is easy to compute such a generator when $q$ is known, but seems hard otherwise.