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?

  • $\begingroup$ What property are you after when a member encrypts a plaintext / ciphertext? Usually these schemes are solved using signatures (or a combination of signatures and encryption). $\endgroup$ – Maarten Bodewes Apr 29 '17 at 10:27
  • $\begingroup$ I know that normally signatures are used, but they do not have the correct properties, because the signature is a lot smaller then the data which means a misbehaving member would just store the signature and discard the data and when asked for it fetch the data from somewhere else meaning replication can no longer be guaranteed. The properties I want is a unique copies of the original data where each can only be created when multiple members sharing a secret act work together, but everybody can recreate the original data from this unique copy $\endgroup$ – Markus Knecht Apr 29 '17 at 15:31

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