How to compute accumulated values in bilinear map accumulators

Question

How to compute $ g^{1/(e_1+s)}$, where $g$ is the generator of group $\mathbb G$, and $e_1$ and $s$ are keys? I know only $s$ and $g^{e_1}$, not $e_1$. $\mathbb G$ has prime order for some prime $p$ and $e_1$ and $s$ are some elements of $\mathbb{Z}_p$.

Answer 1

In your setting this is assumed to be hard. It is exactly the task of producing a forgery for message $s$ of the weakly secure Boneh-Boyen signature scheme (Sec. 3.1) under public key $g^{e_1}$ (note that the scheme is presented in the asymmetric setting but can equally be instantiated in the symmetric setting under the $q$-SDH assumption).

In other words, if you can do this, you can break the $1$-SDH problem.