Suppose I have a group $G$ of unknown order $n$ where $n=p^k\cdot s$, $\gcd(p,s)=1$, $p$ is a known prime, $k,s$ are unknown positive integers and $k,s\ge1$. (Known - $p$ and $p\mid n$, Unknown - $n,k,s$). Assume that it is easy to solve the discrete log problem in the subgroup of order $p$.
I know that if I have an upper bound on $n$, I can use Baby step-giant step to solve the discrete log in $G$. Does Pohlig-Hellman also work if you know the upper bound?
Can I solve the discrete log problem in $G$ using the Pohlig-Hellman algorithm or any other algorithm that has square root complexity in the above group setting?
Can one find $k$ using any of the discrete log solving algorithms?
My probable answers
Assuming that Pohlig-Hellman only works if you precisely know the group order, then no I can't solve the discrete log problem in $G$ as I don't know $n$(or $k$ for that matter).
Not sure what the answer is if I use baby step-giant step but I think you can not find $k$ using Pohlig-Hellman.
Need help in filling the gaps and verifying my answers.