The $n$-strong Diffie Hellman assumption state that given the subset $\{g, g^s,\cdots,g^{s^n}\} \subseteq \mathbb{G}$ in a cyclic group $\mathbb{G}$ of prime order $p$, a PPT algorithm cannot output $g^{\frac{1}{s+\alpha}}$ for any $\alpha \in \mathbb{F}_p$ except with negligible probability.
Does it somehow imply that no PPT algorithm can output an irreducible polynomial $f(X)\in \mathbb{F}_p[X]$ and the element $g^{\frac{1}{f(s)}}$? Or does that entail a strictly stronger assumption?