# Secret Sharing Scheme, with joining players, no central entity

I am facing the the following problem: Lets assume i have a secret, $S$, I want to share it to $N$ players so that $K$ of them can cooperate in order to obtain the secret ($K$,$N$ scheme). Is there a scheme that supports the joining of a new player where a set of $K$ players can issue a new key without obtaining the secret itself resulting a $(K,N+1)$ scheme and with no central entity meaning there is no dealer after the first dealership.

This is a classic problem that can be solved with secure multiparty computation. When a new player joins, the $K$ parties who approve them joining run a secure computation protocol to generate their share. Concretely, if Shamir sharing is used, then the function computed is "reconstruct the polynomial $p$ from the $K$ shares input, and output the share $p(\alpha_{N+1})$, where $\alpha_1,\ldots,\alpha_{N+1}$ are field elements and $\alpha_i$ is associated with party $P_i$.