For a thought experiment I'm looking for a threshold encryption scheme where $k$ out of $n$ shares are needed to encrypt the data and everybody can decrypt the data. However, I only found schemes needing $k$ out of $n$ for decryption. I know that it would be an uncommon scenario.
The problem behind this is that I would have multiple groups and some data and I want to store that data in a replicated way and be able to ensure the replication.
An assumption is that more than $k$ member in a group $G_i$ are honest and follow the protocol. Now if I provide a piece of data, then based of its hash $r$ (replication factor) groups are selected and 1 member per group $G_i$ is selected. The group then collaborates to encrypt the data and the selected member stores it.
The above would be performed to prevent the member $M_j$ from just reading another replica's state and returning it when asked for it instead of actually storing it. The group member's would have a way to see if a storage request is valid and was not already executed and would not re-encrypt the data if they already have or if its invalid.
Additional Information After some further research I think what I'm looking for is a message recovery threshold signature scheme. But my search did not yield Any results. Does anybody know about such a scheme?